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Theorem tz7.48-2 7582
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 3657 . . 3 On ⊆ On
2 onelon 5786 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
32ancoms 468 . . . . . . . 8 ((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On)
4 tz7.48.1 . . . . . . . . . . 11 𝐹 Fn On
5 fndm 6028 . . . . . . . . . . 11 (𝐹 Fn On → dom 𝐹 = On)
64, 5ax-mp 5 . . . . . . . . . 10 dom 𝐹 = On
76eleq2i 2722 . . . . . . . . 9 (𝑦 ∈ dom 𝐹𝑦 ∈ On)
8 fnfun 6026 . . . . . . . . . . . . 13 (𝐹 Fn On → Fun 𝐹)
94, 8ax-mp 5 . . . . . . . . . . . 12 Fun 𝐹
10 funfvima 6532 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
119, 10mpan 706 . . . . . . . . . . 11 (𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
1211impcom 445 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ (𝐹𝑥))
13 eleq1a 2725 . . . . . . . . . . 11 ((𝐹𝑦) ∈ (𝐹𝑥) → ((𝐹𝑥) = (𝐹𝑦) → (𝐹𝑥) ∈ (𝐹𝑥)))
14 eldifn 3766 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) ∈ (𝐹𝑥))
1513, 14nsyli 155 . . . . . . . . . 10 ((𝐹𝑦) ∈ (𝐹𝑥) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
1612, 15syl 17 . . . . . . . . 9 ((𝑦𝑥𝑦 ∈ dom 𝐹) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
177, 16sylan2br 492 . . . . . . . 8 ((𝑦𝑥𝑦 ∈ On) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
183, 17syldan 486 . . . . . . 7 ((𝑦𝑥𝑥 ∈ On) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
1918expimpd 628 . . . . . 6 (𝑦𝑥 → ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ¬ (𝐹𝑥) = (𝐹𝑦)))
2019com12 32 . . . . 5 ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
2120ralrimiv 2994 . . . 4 ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦))
2221ralimiaa 2980 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦))
234tz7.48lem 7581 . . 3 ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹 ↾ On))
241, 22, 23sylancr 696 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun (𝐹 ↾ On))
25 fnrel 6027 . . . . . 6 (𝐹 Fn On → Rel 𝐹)
264, 25ax-mp 5 . . . . 5 Rel 𝐹
276eqimssi 3692 . . . . 5 dom 𝐹 ⊆ On
28 relssres 5472 . . . . 5 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
2926, 27, 28mp2an 708 . . . 4 (𝐹 ↾ On) = 𝐹
3029cnveqi 5329 . . 3 (𝐹 ↾ On) = 𝐹
3130funeqi 5947 . 2 (Fun (𝐹 ↾ On) ↔ Fun 𝐹)
3224, 31sylib 208 1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  cdif 3604  wss 3607  ccnv 5142  dom cdm 5143  cres 5145  cima 5146  Rel wrel 5148  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  cfv 5926
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-pr 4936
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-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  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-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934
This theorem is referenced by:  tz7.48-3  7584
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