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Theorem hashdom 13206
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 12811 . . . . . . . 8 (1...((#‘𝐵) − (#‘𝐴))) ∈ Fin
2 ficardom 8825 . . . . . . . 8 ((1...((#‘𝐵) − (#‘𝐴))) ∈ Fin → (card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω)
31, 2ax-mp 5 . . . . . . 7 (card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω
4 eqid 2651 . . . . . . . . . . . . . 14 (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
54hashgval 13160 . . . . . . . . . . . . 13 (𝐴 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (#‘𝐴))
65ad2antrr 762 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (#‘𝐴))
74hashgval 13160 . . . . . . . . . . . . . 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 13185 . . . . . . . . . . . . . . . 16 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
109ad2antrr 762 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐴) ∈ ℕ0)
11 hashcl 13185 . . . . . . . . . . . . . . . 16 (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ0)
1211ad2antlr 763 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐵) ∈ ℕ0)
13 simpr 476 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐴) ≤ (#‘𝐵))
14 nn0sub2 11476 . . . . . . . . . . . . . . 15 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((#‘𝐵) − (#‘𝐴)) ∈ ℕ0)
1510, 12, 13, 14syl3anc 1366 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((#‘𝐵) − (#‘𝐴)) ∈ ℕ0)
16 hashfz1 13174 . . . . . . . . . . . . . 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 2697 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((#‘𝐵) − (#‘𝐴))))) = ((#‘𝐵) − (#‘𝐴)))
196, 18oveq12d 6708 . . . . . . . . . . 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 11391 . . . . . . . . . . . . 13 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℂ)
2111nn0cnd 11391 . . . . . . . . . . . . 13 (𝐵 ∈ Fin → (#‘𝐵) ∈ ℂ)
22 pncan3 10327 . . . . . . . . . . . . 13 (((#‘𝐴) ∈ ℂ ∧ (#‘𝐵) ∈ ℂ) → ((#‘𝐴) + ((#‘𝐵) − (#‘𝐴))) = (#‘𝐵))
2320, 21, 22syl2an 493 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) + ((#‘𝐵) − (#‘𝐴))) = (#‘𝐵))
2423adantr 480 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((#‘𝐴) + ((#‘𝐵) − (#‘𝐴))) = (#‘𝐵))
2519, 24eqtrd 2685 . . . . . . . . . 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 8825 . . . . . . . . . . . 12 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
2726ad2antrr 762 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (card‘𝐴) ∈ ω)
284hashgadd 13204 . . . . . . . . . . 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 695 . . . . . . . . . 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 13160 . . . . . . . . . . 11 (𝐵 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (#‘𝐵))
3130ad2antlr 763 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (#‘𝐵))
3225, 29, 313eqtr4d 2695 . . . . . . . . 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 6233 . . . . . . . 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 12810 . . . . . . . . 9 (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0
35 nnacl 7736 . . . . . . . . . 10 (((card‘𝐴) ∈ ω ∧ (card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω) → ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) ∈ ω)
3627, 3, 35sylancl 695 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) ∈ ω)
37 f1ocnvfv1 6572 . . . . . . . . 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 696 . . . . . . . 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 8825 . . . . . . . . . 10 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
4039ad2antlr 763 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (card‘𝐵) ∈ ω)
41 f1ocnvfv1 6572 . . . . . . . . 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 696 . . . . . . . 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 2693 . . . . . . 7 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) = (card‘𝐵))
44 oveq2 6698 . . . . . . . . 9 (𝑦 = (card‘(1...((#‘𝐵) − (#‘𝐴)))) → ((card‘𝐴) +𝑜 𝑦) = ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))))
4544eqeq1d 2653 . . . . . . . 8 (𝑦 = (card‘(1...((#‘𝐵) − (#‘𝐴)))) → (((card‘𝐴) +𝑜 𝑦) = (card‘𝐵) ↔ ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) = (card‘𝐵)))
4645rspcev 3340 . . . . . . 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 696 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵))
4847ex 449 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) → ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
49 cardnn 8827 . . . . . . . . . 10 (𝑦 ∈ ω → (card‘𝑦) = 𝑦)
5049adantl 481 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (card‘𝑦) = 𝑦)
5150oveq2d 6706 . . . . . . . 8 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → ((card‘𝐴) +𝑜 (card‘𝑦)) = ((card‘𝐴) +𝑜 𝑦))
5251eqeq1d 2653 . . . . . . 7 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +𝑜 (card‘𝑦)) = (card‘𝐵) ↔ ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
53 fveq2 6229 . . . . . . . 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 8194 . . . . . . . . 9 (𝑦 ∈ ω → 𝑦 ∈ Fin)
55 ficardom 8825 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
564hashgadd 13204 . . . . . . . . . . . . . 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 493 . . . . . . . . . . . . 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 13160 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦)) = (#‘𝑦))
595, 58oveqan12d 6709 . . . . . . . . . . . . 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 2685 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = ((#‘𝐴) + (#‘𝑦)))
6160adantlr 751 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = ((#‘𝐴) + (#‘𝑦)))
6230ad2antlr 763 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (#‘𝐵))
6361, 62eqeq12d 2666 . . . . . . . . . 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 13185 . . . . . . . . . . . . . . 15 (𝑦 ∈ Fin → (#‘𝑦) ∈ ℕ0)
6564nn0ge0d 11392 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin → 0 ≤ (#‘𝑦))
6665adantl 481 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → 0 ≤ (#‘𝑦))
679nn0red 11390 . . . . . . . . . . . . . 14 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℝ)
6864nn0red 11390 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin → (#‘𝑦) ∈ ℝ)
69 addge01 10576 . . . . . . . . . . . . . 14 (((#‘𝐴) ∈ ℝ ∧ (#‘𝑦) ∈ ℝ) → (0 ≤ (#‘𝑦) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦))))
7067, 68, 69syl2an 493 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (0 ≤ (#‘𝑦) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦))))
7166, 70mpbid 222 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦)))
7271adantlr 751 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦)))
73 breq2 4689 . . . . . . . . . . 11 (((#‘𝐴) + (#‘𝑦)) = (#‘𝐵) → ((#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦)) ↔ (#‘𝐴) ≤ (#‘𝐵)))
7472, 73syl5ibcom 235 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((#‘𝐴) + (#‘𝑦)) = (#‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
7563, 74sylbid 230 . . . . . . . . 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 490 . . . . . . . 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 34 . . . . . . 7 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +𝑜 (card‘𝑦)) = (card‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
7852, 77sylbird 250 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +𝑜 𝑦) = (card‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
7978rexlimdva 3060 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
8048, 79impbid 202 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
81 nnawordex 7762 . . . . 5 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
8226, 39, 81syl2an 493 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
83 finnum 8812 . . . . 5 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
84 finnum 8812 . . . . 5 (𝐵 ∈ Fin → 𝐵 ∈ dom card)
85 carddom2 8841 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))
8683, 84, 85syl2an 493 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))
8780, 82, 863bitr2d 296 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴𝐵))
8887adantlr 751 . 2 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴𝐵))
89 hashxrcl 13186 . . . . . 6 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℝ*)
9089ad2antrr 762 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) ∈ ℝ*)
91 pnfge 12002 . . . . 5 ((#‘𝐴) ∈ ℝ* → (#‘𝐴) ≤ +∞)
9290, 91syl 17 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) ≤ +∞)
93 hashinf 13162 . . . . 5 ((𝐵𝑉 ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞)
9493adantll 750 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞)
9592, 94breqtrrd 4713 . . 3 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) ≤ (#‘𝐵))
96 isinffi 8856 . . . . . 6 ((¬ 𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓:𝐴1-1𝐵)
9796ancoms 468 . . . . 5 ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴1-1𝐵)
9897adantlr 751 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴1-1𝐵)
99 brdomg 8007 . . . . 5 (𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
10099ad2antlr 763 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
10198, 100mpbird 247 . . 3 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → 𝐴𝐵)
10295, 1012thd 255 . 2 (((𝐴 ∈ Fin ∧ 𝐵𝑉) ∧ ¬ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴𝐵))
10388, 102pm2.61dan 849 1 ((𝐴 ∈ Fin ∧ 𝐵𝑉) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  Vcvv 3231  wss 3607   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  cres 5145  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  ωcom 7107  reccrdg 7550   +𝑜 coa 7602  cdom 7995  Fincfn 7997  cardccrd 8799  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  +∞cpnf 10109  *cxr 10111  cle 10113  cmin 10304  0cn0 11330  ...cfz 12364  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158
This theorem is referenced by:  hashdomi  13207  hashsdom  13208  hashun2  13210  hashss  13235  hashsslei  13251  hashfun  13262  hashf1  13279  hashge3el3dif  13306  isercoll  14442  phicl2  15520  phibnd  15523  prmreclem2  15668  prmreclem3  15669  4sqlem11  15706  vdwlem11  15742  ramub2  15765  0ram  15771  ram0  15773  sylow1lem4  18062  pgpssslw  18075  fislw  18086  znfld  19957  znidomb  19958  fta1blem  23973  birthdaylem3  24725  basellem4  24855  ppiwordi  24933  musum  24962  ppiub  24974  chpub  24990  lgsqrlem4  25119  upgrex  26032  sizusglecusg  26415  derangenlem  31279  subfaclefac  31284  erdsze2lem1  31311  snmlff  31437  idomsubgmo  38093  aacllem  42875
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