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Theorem hashdom 12981
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 ∧ 𝐵𝑉) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴𝐵))

Proof of Theorem hashdom
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfi 12588 . . . . . . . 8 (1...((#‘𝐵) − (#‘𝐴))) ∈ Fin
2 ficardom 8647 . . . . . . . 8 ((1...((#‘𝐵) − (#‘𝐴))) ∈ Fin → (card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω)
31, 2ax-mp 5 . . . . . . 7 (card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω
4 eqid 2609 . . . . . . . . . . . . . 14 (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
54hashgval 12937 . . . . . . . . . . . . 13 (𝐴 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (#‘𝐴))
65ad2antrr 757 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (#‘𝐴))
74hashgval 12937 . . . . . . . . . . . . . 14 ((1...((#‘𝐵) − (#‘𝐴))) ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((#‘𝐵) − (#‘𝐴))))) = (#‘(1...((#‘𝐵) − (#‘𝐴)))))
81, 7ax-mp 5 . . . . . . . . . . . . 13 ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((#‘𝐵) − (#‘𝐴))))) = (#‘(1...((#‘𝐵) − (#‘𝐴))))
9 hashcl 12961 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
109ad2antrr 757 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐴) ∈ ℕ0)
11 hashcl 12961 . . . . . . . . . . . . . . . 16 (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ0)
1211ad2antlr 758 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐵) ∈ ℕ0)
13 simpr 475 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐴) ≤ (#‘𝐵))
14 nn0sub2 11271 . . . . . . . . . . . . . . 15 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((#‘𝐵) − (#‘𝐴)) ∈ ℕ0)
1510, 12, 13, 14syl3anc 1317 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((#‘𝐵) − (#‘𝐴)) ∈ ℕ0)
16 hashfz1 12948 . . . . . . . . . . . . . 14 (((#‘𝐵) − (#‘𝐴)) ∈ ℕ0 → (#‘(1...((#‘𝐵) − (#‘𝐴)))) = ((#‘𝐵) − (#‘𝐴)))
1715, 16syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘(1...((#‘𝐵) − (#‘𝐴)))) = ((#‘𝐵) − (#‘𝐴)))
188, 17syl5eq 2655 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((#‘𝐵) − (#‘𝐴))))) = ((#‘𝐵) − (#‘𝐴)))
196, 18oveq12d 6545 . . . . . . . . . . 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))))
209nn0cnd 11200 . . . . . . . . . . . . 13 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℂ)
2111nn0cnd 11200 . . . . . . . . . . . . 13 (𝐵 ∈ Fin → (#‘𝐵) ∈ ℂ)
22 pncan3 10140 . . . . . . . . . . . . 13 (((#‘𝐴) ∈ ℂ ∧ (#‘𝐵) ∈ ℂ) → ((#‘𝐴) + ((#‘𝐵) − (#‘𝐴))) = (#‘𝐵))
2320, 21, 22syl2an 492 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) + ((#‘𝐵) − (#‘𝐴))) = (#‘𝐵))
2423adantr 479 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((#‘𝐴) + ((#‘𝐵) − (#‘𝐴))) = (#‘𝐵))
2519, 24eqtrd 2643 . . . . . . . . . 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 8647 . . . . . . . . . . . 12 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
2726ad2antrr 757 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (card‘𝐴) ∈ ω)
284hashgadd 12979 . . . . . . . . . . 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)))))))
2927, 3, 28sylancl 692 . . . . . . . . . 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)))))))
304hashgval 12937 . . . . . . . . . . 11 (𝐵 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (#‘𝐵))
3130ad2antlr 758 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (#‘𝐵))
3225, 29, 313eqtr4d 2653 . . . . . . . . 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)))
3332fveq2d 6092 . . . . . . . 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))))
344hashgf1o 12587 . . . . . . . . 9 (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0
35 nnacl 7555 . . . . . . . . . 10 (((card‘𝐴) ∈ ω ∧ (card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω) → ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) ∈ ω)
3627, 3, 35sylancl 692 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) ∈ ω)
37 f1ocnvfv1 6410 . . . . . . . . 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))))))
3834, 36, 37sylancr 693 . . . . . . . 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 8647 . . . . . . . . . 10 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
4039ad2antlr 758 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (card‘𝐵) ∈ ω)
41 f1ocnvfv1 6410 . . . . . . . . 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))
4234, 40, 41sylancr 693 . . . . . . . 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))
4333, 38, 423eqtr3d 2651 . . . . . . 7 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) = (card‘𝐵))
44 oveq2 6535 . . . . . . . . 9 (𝑦 = (card‘(1...((#‘𝐵) − (#‘𝐴)))) → ((card‘𝐴) +𝑜 𝑦) = ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))))
4544eqeq1d 2611 . . . . . . . 8 (𝑦 = (card‘(1...((#‘𝐵) − (#‘𝐴)))) → (((card‘𝐴) +𝑜 𝑦) = (card‘𝐵) ↔ ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) = (card‘𝐵)))
4645rspcev 3281 . . . . . . 7 (((card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω ∧ ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) = (card‘𝐵)) → ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵))
473, 43, 46sylancr 693 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵))
4847ex 448 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) → ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
49 cardnn 8649 . . . . . . . . . 10 (𝑦 ∈ ω → (card‘𝑦) = 𝑦)
5049adantl 480 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (card‘𝑦) = 𝑦)
5150oveq2d 6543 . . . . . . . 8 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → ((card‘𝐴) +𝑜 (card‘𝑦)) = ((card‘𝐴) +𝑜 𝑦))
5251eqeq1d 2611 . . . . . . 7 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +𝑜 (card‘𝑦)) = (card‘𝐵) ↔ ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
53 fveq2 6088 . . . . . . . 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 8015 . . . . . . . . 9 (𝑦 ∈ ω → 𝑦 ∈ Fin)
55 ficardom 8647 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
564hashgadd 12979 . . . . . . . . . . . . . 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))))
5726, 55, 56syl2an 492 . . . . . . . . . . . . 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))))
584hashgval 12937 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦)) = (#‘𝑦))
595, 58oveqan12d 6546 . . . . . . . . . . . . 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)))
6057, 59eqtrd 2643 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = ((#‘𝐴) + (#‘𝑦)))
6160adantlr 746 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = ((#‘𝐴) + (#‘𝑦)))
6230ad2antlr 758 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (#‘𝐵))
6361, 62eqeq12d 2624 . . . . . . . . . 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 12961 . . . . . . . . . . . . . . 15 (𝑦 ∈ Fin → (#‘𝑦) ∈ ℕ0)
6564nn0ge0d 11201 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin → 0 ≤ (#‘𝑦))
6665adantl 480 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → 0 ≤ (#‘𝑦))
679nn0red 11199 . . . . . . . . . . . . . 14 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℝ)
6864nn0red 11199 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin → (#‘𝑦) ∈ ℝ)
69 addge01 10387 . . . . . . . . . . . . . 14 (((#‘𝐴) ∈ ℝ ∧ (#‘𝑦) ∈ ℝ) → (0 ≤ (#‘𝑦) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦))))
7067, 68, 69syl2an 492 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (0 ≤ (#‘𝑦) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦))))
7166, 70mpbid 220 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦)))
7271adantlr 746 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦)))
73 breq2 4581 . . . . . . . . . . 11 (((#‘𝐴) + (#‘𝑦)) = (#‘𝐵) → ((#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦)) ↔ (#‘𝐴) ≤ (#‘𝐵)))
7472, 73syl5ibcom 233 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((#‘𝐴) + (#‘𝑦)) = (#‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
7563, 74sylbid 228 . . . . . . . . 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)))
7654, 75sylan2 489 . . . . . . . 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)))
7753, 76syl5 33 . . . . . . 7 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +𝑜 (card‘𝑦)) = (card‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
7852, 77sylbird 248 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +𝑜 𝑦) = (card‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
7978rexlimdva 3012 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
8048, 79impbid 200 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
81 nnawordex 7581 . . . . 5 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
8226, 39, 81syl2an 492 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
83 finnum 8634 . . . . 5 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
84 finnum 8634 . . . . 5 (𝐵 ∈ Fin → 𝐵 ∈ dom card)
85 carddom2 8663 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))
8683, 84, 85syl2an 492 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))
8780, 82, 863bitr2d 294 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴𝐵))
8887adantlr 746 . 2 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴𝐵))
89 hashxrcl 12962 . . . . . 6 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℝ*)
9089ad2antrr 757 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) ∈ ℝ*)
91 pnfge 11801 . . . . 5 ((#‘𝐴) ∈ ℝ* → (#‘𝐴) ≤ +∞)
9290, 91syl 17 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) ≤ +∞)
93 hashinf 12939 . . . . 5 ((𝐵𝑉 ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞)
9493adantll 745 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞)
9592, 94breqtrrd 4605 . . 3 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) ≤ (#‘𝐵))
96 isinffi 8678 . . . . . 6 ((¬ 𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓:𝐴1-1𝐵)
9796ancoms 467 . . . . 5 ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴1-1𝐵)
9897adantlr 746 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴1-1𝐵)
99 brdomg 7828 . . . . 5 (𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
10099ad2antlr 758 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
10198, 100mpbird 245 . . 3 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → 𝐴𝐵)
10295, 1012thd 253 . 2 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴𝐵))
10388, 102pm2.61dan 827 1 ((𝐴 ∈ Fin ∧ 𝐵𝑉) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1976  wrex 2896  Vcvv 3172  wss 3539   class class class wbr 4577  cmpt 4637  ccnv 5027  dom cdm 5028  cres 5030  1-1wf1 5787  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  ωcom 6934  reccrdg 7369   +𝑜 coa 7421  cdom 7816  Fincfn 7818  cardccrd 8621  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795  +∞cpnf 9927  *cxr 9929  cle 9931  cmin 10117  0cn0 11139  ...cfz 12152  #chash 12934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-hash 12935
This theorem is referenced by:  hashdomi  12982  hashsdom  12983  hashun2  12985  hashss  13010  hashsslei  13025  hashfun  13036  hashf1  13050  hashge3el3dif  13072  isercoll  14192  phicl2  15257  phibnd  15260  prmreclem2  15405  prmreclem3  15406  4sqlem11  15443  vdwlem11  15479  ramub2  15502  0ram  15508  ram0  15510  sylow1lem4  17785  pgpssslw  17798  fislw  17809  znfld  19673  znidomb  19674  fta1blem  23649  birthdaylem3  24397  basellem4  24527  ppiwordi  24605  musum  24634  ppiub  24646  chpub  24662  lgsqrlem4  24791  umgraex  25618  sizeusglecusg  25780  konigsberg  26280  derangenlem  30213  subfaclefac  30218  erdsze2lem1  30245  snmlff  30371  idomsubgmo  36598  upgrex  40320  sizusglecusg  40681  aacllem  42319
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