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Theorem tz7.49c 7486
 Description: Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)
Hypothesis
Ref Expression
tz7.49c.1 𝐹 Fn On
Assertion
Ref Expression
tz7.49c ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem tz7.49c
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 tz7.49c.1 . . 3 𝐹 Fn On
2 biid 251 . . 3 (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
31, 2tz7.49 7485 . 2 ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
4 3simpc 1058 . . . 4 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → ((𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
5 onss 6937 . . . . . . . . 9 (𝑥 ∈ On → 𝑥 ⊆ On)
6 fnssres 5962 . . . . . . . . 9 ((𝐹 Fn On ∧ 𝑥 ⊆ On) → (𝐹𝑥) Fn 𝑥)
71, 5, 6sylancr 694 . . . . . . . 8 (𝑥 ∈ On → (𝐹𝑥) Fn 𝑥)
8 df-ima 5087 . . . . . . . . . 10 (𝐹𝑥) = ran (𝐹𝑥)
98eqeq1i 2626 . . . . . . . . 9 ((𝐹𝑥) = 𝐴 ↔ ran (𝐹𝑥) = 𝐴)
109biimpi 206 . . . . . . . 8 ((𝐹𝑥) = 𝐴 → ran (𝐹𝑥) = 𝐴)
117, 10anim12i 589 . . . . . . 7 ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) → ((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴))
1211anim1i 591 . . . . . 6 (((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)) → (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
13 dff1o2 6099 . . . . . . 7 ((𝐹𝑥):𝑥1-1-onto𝐴 ↔ ((𝐹𝑥) Fn 𝑥 ∧ Fun (𝐹𝑥) ∧ ran (𝐹𝑥) = 𝐴))
14 3anan32 1048 . . . . . . 7 (((𝐹𝑥) Fn 𝑥 ∧ Fun (𝐹𝑥) ∧ ran (𝐹𝑥) = 𝐴) ↔ (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
1513, 14bitri 264 . . . . . 6 ((𝐹𝑥):𝑥1-1-onto𝐴 ↔ (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
1612, 15sylibr 224 . . . . 5 (((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)) → (𝐹𝑥):𝑥1-1-onto𝐴)
1716expl 647 . . . 4 (𝑥 ∈ On → (((𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → (𝐹𝑥):𝑥1-1-onto𝐴))
184, 17syl5 34 . . 3 (𝑥 ∈ On → ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → (𝐹𝑥):𝑥1-1-onto𝐴))
1918reximia 3003 . 2 (∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
203, 19syl 17 1 ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908   ∖ cdif 3552   ⊆ wss 3555  ∅c0 3891  ◡ccnv 5073  ran crn 5075   ↾ cres 5076   “ cima 5077  Oncon0 5682  Fun wfun 5841   Fn wfn 5842  –1-1-onto→wf1o 5846  ‘cfv 5847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855 This theorem is referenced by:  dfac8alem  8796  dnnumch1  37094
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