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Theorem hashdom 14389

Description: Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.)

**Assertion**
Ref	Expression
hashdom	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))

Proof of Theorem hashdom Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfi 13982	. . . . . . . 8 ⊢ (1...((♯‘𝐵) − (♯‘𝐴))) ∈ Fin
2		ficardom 9916	. . . . . . . 8 ⊢ ((1...((♯‘𝐵) − (♯‘𝐴))) ∈ Fin → (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω)
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω
4		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
5	4	hashgval 14343	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (♯‘𝐴))
6	5	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (♯‘𝐴))
7	4	hashgval 14343	. . . . . . . . . . . . . 14 ⊢ ((1...((♯‘𝐵) − (♯‘𝐴))) ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (♯‘(1...((♯‘𝐵) − (♯‘𝐴)))))
8	1, 7	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (♯‘(1...((♯‘𝐵) − (♯‘𝐴))))
9		hashcl 14366	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
10	9	ad2antrr 736	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐴) ∈ ℕ₀)
11		hashcl 14366	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
12	11	ad2antlr 737	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐵) ∈ ℕ₀)
13		simpr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐴) ≤ (♯‘𝐵))
14		nn0sub2 12631	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀ ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℕ₀)
15	10, 12, 13, 14	syl3anc 1389	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℕ₀)
16		hashfz1 14356	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐵) − (♯‘𝐴)) ∈ ℕ₀ → (♯‘(1...((♯‘𝐵) − (♯‘𝐴)))) = ((♯‘𝐵) − (♯‘𝐴)))
17	15, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘(1...((♯‘𝐵) − (♯‘𝐴)))) = ((♯‘𝐵) − (♯‘𝐴)))
18	8, 17	eqtrid 2808	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = ((♯‘𝐵) − (♯‘𝐴)))
19	6, 18	oveq12d 7410	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴)))))) = ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))))
20	9	nn0cnd 12541	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
21	11	nn0cnd 12541	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℂ)
22		pncan3 11435	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐴) ∈ ℂ ∧ (♯‘𝐵) ∈ ℂ) → ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))) = (♯‘𝐵))
23	20, 21, 22	syl2an 605	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))) = (♯‘𝐵))
24	23	adantr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))) = (♯‘𝐵))
25	19, 24	eqtrd 2796	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴)))))) = (♯‘𝐵))
26		ficardom 9916	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
27	26	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (card‘𝐴) ∈ ω)
28	4	hashgadd 14387	. . . . . . . . . . 11 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴)))))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴)))))))
29	27, 3, 28	sylancl 595	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴)))))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴)))))))
30	4	hashgval 14343	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
31	30	ad2antlr 737	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
32	25, 29, 31	3eqtr4d 2806	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴)))))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)))
33	32	fveq2d 6867	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))) = (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵))))
34	4	hashgf1o 13981	. . . . . . . . 9 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ₀
35		nnacl 8576	. . . . . . . . . 10 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) ∈ ω)
36	27, 3, 35	sylancl 595	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) ∈ ω)
37		f1ocnvfv1 7256	. . . . . . . . 9 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ₀ ∧ ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) ∈ ω) → (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))) = ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))
38	34, 36, 37	sylancr 596	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))) = ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))
39		ficardom 9916	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
40	39	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (card‘𝐵) ∈ ω)
41		f1ocnvfv1 7256	. . . . . . . . 9 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ₀ ∧ (card‘𝐵) ∈ ω) → (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵))) = (card‘𝐵))
42	34, 40, 41	sylancr 596	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵))) = (card‘𝐵))
43	33, 38, 42	3eqtr3d 2804	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (card‘𝐵))
44		oveq2 7400	. . . . . . . . 9 ⊢ (𝑦 = (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) → ((card‘𝐴) +_o 𝑦) = ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))
45	44	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑦 = (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) → (((card‘𝐴) +_o 𝑦) = (card‘𝐵) ↔ ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (card‘𝐵)))
46	45	rspcev 3581	. . . . . . 7 ⊢ (((card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω ∧ ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (card‘𝐵)) → ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵))
47	3, 43, 46	sylancr 596	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵))
48	47	ex 416	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) → ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
49		cardnn 9918	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (card‘𝑦) = 𝑦)
50	49	adantl 485	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (card‘𝑦) = 𝑦)
51	50	oveq2d 7408	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → ((card‘𝐴) +_o (card‘𝑦)) = ((card‘𝐴) +_o 𝑦))
52	51	eqeq1d 2763	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o (card‘𝑦)) = (card‘𝐵) ↔ ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
53		fveq2 6863	. . . . . . . 8 ⊢ (((card‘𝐴) +_o (card‘𝑦)) = (card‘𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)))
54		nnfi 9132	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → 𝑦 ∈ Fin)
55		ficardom 9916	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
56	4	hashgadd 14387	. . . . . . . . . . . . . 14 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝑦) ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦))))
57	26, 55, 56	syl2an 605	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦))))
58	4	hashgval 14343	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦)) = (♯‘𝑦))
59	5, 58	oveqan12d 7411	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦))) = ((♯‘𝐴) + (♯‘𝑦)))
60	57, 59	eqtrd 2796	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((♯‘𝐴) + (♯‘𝑦)))
61	60	adantlr 725	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((♯‘𝐴) + (♯‘𝑦)))
62	30	ad2antlr 737	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
63	61, 62	eqeq12d 2777	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) ↔ ((♯‘𝐴) + (♯‘𝑦)) = (♯‘𝐵)))
64		hashcl 14366	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
65	64	nn0ge0d 12542	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → 0 ≤ (♯‘𝑦))
66	65	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → 0 ≤ (♯‘𝑦))
67	9	nn0red 12540	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ)
68	64	nn0red 12540	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℝ)
69		addge01 11694	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐴) ∈ ℝ ∧ (♯‘𝑦) ∈ ℝ) → (0 ≤ (♯‘𝑦) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦))))
70	67, 68, 69	syl2an 605	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (0 ≤ (♯‘𝑦) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦))))
71	66, 70	mpbid 234	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)))
72	71	adantlr 725	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)))
73		breq2 5103	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) + (♯‘𝑦)) = (♯‘𝐵) → ((♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)) ↔ (♯‘𝐴) ≤ (♯‘𝐵)))
74	72, 73	syl5ibcom 247	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((♯‘𝐴) + (♯‘𝑦)) = (♯‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
75	63, 74	sylbid 242	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) → (♯‘𝐴) ≤ (♯‘𝐵)))
76	54, 75	sylan2 602	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) → (♯‘𝐴) ≤ (♯‘𝐵)))
77	53, 76	syl5 34	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o (card‘𝑦)) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
78	52, 77	sylbird 262	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o 𝑦) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
79	78	rexlimdva 3162	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
80	48, 79	impbid 214	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
81		nnawordex 8602	. . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
82	26, 39, 81	syl2an 605	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
83		finnum 9903	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
84		finnum 9903	. . . . 5 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card)
85		carddom2 9932	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
86	83, 84, 85	syl2an 605	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
87	80, 82, 86	3bitr2d 309	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
88	87	adantlr 725	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
89		hashxrcl 14367	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ^*)
90	89	ad2antrr 736	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ∈ ℝ^*)
91		pnfge 13129	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℝ^* → (♯‘𝐴) ≤ +∞)
92	90, 91	syl 17	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ≤ +∞)
93		hashinf 14345	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞)
94	93	adantll 724	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞)
95	92, 94	breqtrrd 5127	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ≤ (♯‘𝐵))
96		isinffi 9947	. . . . . 6 ⊢ ((¬ 𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
97	96	ancoms 462	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
98	97	adantlr 725	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
99		brdomg 8935	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
100	99	ad2antlr 737	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
101	98, 100	mpbird 259	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → 𝐴 ≼ 𝐵)
102	95, 101	2thd 267	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
103	88, 102	pm2.61dan 822	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∃wex 1798 ∈ wcel 2141 ∃wrex 3085 Vcvv 3453 ⊆ wss 3904 class class class wbr 5099 ↦ cmpt 5180 ^◡ccnv 5644 dom cdm 5645 ↾ cres 5647 –1-1→wf1 6514 –1-1-onto→wf1o 6516 ‘cfv 6517 (class class class)co 7392 ωcom 7842 reccrdg 8375 +_o coa 8429 ≼ cdom 8921 Fincfn 8923 cardccrd 9890 ℂcc 11068 ℝcr 11069 0cc0 11070 1c1 11071 + caddc 11073 +∞cpnf 11210 ℝ^*cxr 11212 ≤ cle 11214 − cmin 11411 ℕ₀cn0 12478 ...cfz 13509 ♯chash 14340

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-oadd 8436 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-n0 12479 df-xnn0 12552 df-z 12566 df-uz 12837 df-fz 13510 df-hash 14341

This theorem is referenced by: hashdomi 14390 hashsdom 14391 hashun2 14393 hashss 14419 hashsslei 14436 hashfun 14447 hashf1 14467 hashge3el3dif 14497 isercoll 15678 phicl2 16786 phibnd 16789 prmreclem2 16936 prmreclem3 16937 4sqlem11 16974 vdwlem11 17010 ramub2 17033 0ram 17039 ram0 17041 sylow1lem4 19624 pgpssslw 19637 fislw 19648 znfld 21592 znidomb 21593 fta1blem 26211 birthdaylem3 26995 basellem4 27125 ppiwordi 27203 musum 27232 ppiub 27245 chpub 27261 lgsqrlem4 27390 upgrex 29239 sizusglecusg 29610 derangenlem 35485 subfaclefac 35490 erdsze2lem1 35517 snmlff 35643 hashnexinj 42709 idomsubgmo 43734 aacllem 50386

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