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Theorem tz7.49c 8487
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 261 . . 3 (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
31, 2tz7.49 8486 . 2 ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
4 3simpc 1150 . . . 4 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → ((𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
5 onss 7806 . . . . . . . . 9 (𝑥 ∈ On → 𝑥 ⊆ On)
6 fnssres 6690 . . . . . . . . 9 ((𝐹 Fn On ∧ 𝑥 ⊆ On) → (𝐹𝑥) Fn 𝑥)
71, 5, 6sylancr 587 . . . . . . . 8 (𝑥 ∈ On → (𝐹𝑥) Fn 𝑥)
8 df-ima 5697 . . . . . . . . . 10 (𝐹𝑥) = ran (𝐹𝑥)
98eqeq1i 2741 . . . . . . . . 9 ((𝐹𝑥) = 𝐴 ↔ ran (𝐹𝑥) = 𝐴)
109biimpi 216 . . . . . . . 8 ((𝐹𝑥) = 𝐴 → ran (𝐹𝑥) = 𝐴)
117, 10anim12i 613 . . . . . . 7 ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) → ((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴))
1211anim1i 615 . . . . . 6 (((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)) → (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
13 dff1o2 6852 . . . . . . 7 ((𝐹𝑥):𝑥1-1-onto𝐴 ↔ ((𝐹𝑥) Fn 𝑥 ∧ Fun (𝐹𝑥) ∧ ran (𝐹𝑥) = 𝐴))
14 3anan32 1096 . . . . . . 7 (((𝐹𝑥) Fn 𝑥 ∧ Fun (𝐹𝑥) ∧ ran (𝐹𝑥) = 𝐴) ↔ (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
1513, 14bitri 275 . . . . . 6 ((𝐹𝑥):𝑥1-1-onto𝐴 ↔ (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
1612, 15sylibr 234 . . . . 5 (((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)) → (𝐹𝑥):𝑥1-1-onto𝐴)
1716expl 457 . . . 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 3080 . 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 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  cdif 3947  wss 3950  c0 4332  ccnv 5683  ran crn 5685  cres 5686  cima 5687  Oncon0 6383  Fun wfun 6554   Fn wfn 6555  1-1-ontowf1o 6559  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568
This theorem is referenced by:  dfac8alem  10070  dnnumch1  43061
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