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Theorem tz7.48-2 8444
Description: Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)
Hypothesis
Ref Expression
tz7.48.1 𝐹 Fn On
Assertion
Ref Expression
tz7.48-2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun 𝐹)
Distinct variable group:   𝑥,𝐹
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem tz7.48-2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssid 4004 . . 3 On ⊆ On
2 onelon 6389 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
32ancoms 459 . . . . . . . 8 ((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On)
4 tz7.48.1 . . . . . . . . . . 11 𝐹 Fn On
54fndmi 6653 . . . . . . . . . 10 dom 𝐹 = On
65eleq2i 2825 . . . . . . . . 9 (𝑦 ∈ dom 𝐹𝑦 ∈ On)
7 fnfun 6649 . . . . . . . . . . . . 13 (𝐹 Fn On → Fun 𝐹)
84, 7ax-mp 5 . . . . . . . . . . . 12 Fun 𝐹
9 funfvima 7234 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
108, 9mpan 688 . . . . . . . . . . 11 (𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
1110impcom 408 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ (𝐹𝑥))
12 eleq1a 2828 . . . . . . . . . . 11 ((𝐹𝑦) ∈ (𝐹𝑥) → ((𝐹𝑥) = (𝐹𝑦) → (𝐹𝑥) ∈ (𝐹𝑥)))
13 eldifn 4127 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) ∈ (𝐹𝑥))
1412, 13nsyli 157 . . . . . . . . . 10 ((𝐹𝑦) ∈ (𝐹𝑥) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
1511, 14syl 17 . . . . . . . . 9 ((𝑦𝑥𝑦 ∈ dom 𝐹) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
166, 15sylan2br 595 . . . . . . . 8 ((𝑦𝑥𝑦 ∈ On) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
173, 16syldan 591 . . . . . . 7 ((𝑦𝑥𝑥 ∈ On) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
1817expimpd 454 . . . . . 6 (𝑦𝑥 → ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ¬ (𝐹𝑥) = (𝐹𝑦)))
1918com12 32 . . . . 5 ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
2019ralrimiv 3145 . . . 4 ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦))
2120ralimiaa 3082 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦))
224tz7.48lem 8443 . . 3 ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹 ↾ On))
231, 21, 22sylancr 587 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun (𝐹 ↾ On))
24 fnrel 6651 . . . . . 6 (𝐹 Fn On → Rel 𝐹)
254, 24ax-mp 5 . . . . 5 Rel 𝐹
265eqimssi 4042 . . . . 5 dom 𝐹 ⊆ On
27 relssres 6022 . . . . 5 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
2825, 26, 27mp2an 690 . . . 4 (𝐹 ↾ On) = 𝐹
2928cnveqi 5874 . . 3 (𝐹 ↾ On) = 𝐹
3029funeqi 6569 . 2 (Fun (𝐹 ↾ On) ↔ Fun 𝐹)
3123, 30sylib 217 1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  cdif 3945  wss 3948  ccnv 5675  dom cdm 5676  cres 5678  cima 5679  Rel wrel 5681  Oncon0 6364  Fun wfun 6537   Fn wfn 6538  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551
This theorem is referenced by:  tz7.48-3  8446
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