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Theorem tz7.48-2 8198
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 3937 . . 3 On ⊆ On
2 onelon 6255 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
32ancoms 462 . . . . . . . 8 ((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On)
4 tz7.48.1 . . . . . . . . . . 11 𝐹 Fn On
54fndmi 6500 . . . . . . . . . 10 dom 𝐹 = On
65eleq2i 2830 . . . . . . . . 9 (𝑦 ∈ dom 𝐹𝑦 ∈ On)
7 fnfun 6496 . . . . . . . . . . . . 13 (𝐹 Fn On → Fun 𝐹)
84, 7ax-mp 5 . . . . . . . . . . . 12 Fun 𝐹
9 funfvima 7064 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
108, 9mpan 690 . . . . . . . . . . 11 (𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
1110impcom 411 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ (𝐹𝑥))
12 eleq1a 2834 . . . . . . . . . . 11 ((𝐹𝑦) ∈ (𝐹𝑥) → ((𝐹𝑥) = (𝐹𝑦) → (𝐹𝑥) ∈ (𝐹𝑥)))
13 eldifn 4056 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) ∈ (𝐹𝑥))
1412, 13nsyli 160 . . . . . . . . . 10 ((𝐹𝑦) ∈ (𝐹𝑥) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
1511, 14syl 17 . . . . . . . . 9 ((𝑦𝑥𝑦 ∈ dom 𝐹) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
166, 15sylan2br 598 . . . . . . . 8 ((𝑦𝑥𝑦 ∈ On) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
173, 16syldan 594 . . . . . . 7 ((𝑦𝑥𝑥 ∈ On) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
1817expimpd 457 . . . . . 6 (𝑦𝑥 → ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ¬ (𝐹𝑥) = (𝐹𝑦)))
1918com12 32 . . . . 5 ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
2019ralrimiv 3105 . . . 4 ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦))
2120ralimiaa 3083 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦))
224tz7.48lem 8197 . . 3 ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹 ↾ On))
231, 21, 22sylancr 590 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun (𝐹 ↾ On))
24 fnrel 6498 . . . . . 6 (𝐹 Fn On → Rel 𝐹)
254, 24ax-mp 5 . . . . 5 Rel 𝐹
265eqimssi 3973 . . . . 5 dom 𝐹 ⊆ On
27 relssres 5906 . . . . 5 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
2825, 26, 27mp2an 692 . . . 4 (𝐹 ↾ On) = 𝐹
2928cnveqi 5757 . . 3 (𝐹 ↾ On) = 𝐹
3029funeqi 6418 . 2 (Fun (𝐹 ↾ On) ↔ Fun 𝐹)
3123, 30sylib 221 1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wcel 2111  wral 3062  cdif 3877  wss 3880  ccnv 5564  dom cdm 5565  cres 5567  cima 5568  Rel wrel 5570  Oncon0 6230  Fun wfun 6391   Fn wfn 6392  cfv 6397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-pss 3899  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-tr 5176  df-id 5469  df-eprel 5474  df-po 5482  df-so 5483  df-fr 5523  df-we 5525  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-ord 6233  df-on 6234  df-iota 6355  df-fun 6399  df-fn 6400  df-f 6401  df-f1 6402  df-fv 6405
This theorem is referenced by:  tz7.48-3  8200
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