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Theorem bnj1385 30646
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1385.1 (𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)
bnj1385.2 𝐷 = (dom 𝑓 ∩ dom 𝑔)
bnj1385.3 (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))
bnj1385.4 (𝑥𝐴 → ∀𝑓 𝑥𝐴)
bnj1385.5 (𝜑′ ↔ ∀𝐴 Fun )
bnj1385.6 𝐸 = (dom ∩ dom 𝑔)
bnj1385.7 (𝜓′ ↔ (𝜑′ ∧ ∀𝐴𝑔𝐴 (𝐸) = (𝑔𝐸)))
Assertion
Ref Expression
bnj1385 (𝜓 → Fun 𝐴)
Distinct variable groups:   𝐴,𝑔,,𝑥   𝐷,   𝑓,𝐸   𝑓,𝑔,,𝑥   𝑔,𝜑′
Allowed substitution hints:   𝜑(𝑥,𝑓,𝑔,)   𝜓(𝑥,𝑓,𝑔,)   𝐴(𝑓)   𝐷(𝑥,𝑓,𝑔)   𝐸(𝑥,𝑔,)   𝜑′(𝑥,𝑓,)   𝜓′(𝑥,𝑓,𝑔,)

Proof of Theorem bnj1385
StepHypRef Expression
1 nfv 1840 . . . . . . 7 (𝑓𝐴 → Fun 𝑓)
2 bnj1385.4 . . . . . . . . . 10 (𝑥𝐴 → ∀𝑓 𝑥𝐴)
32nfcii 2752 . . . . . . . . 9 𝑓𝐴
43nfcri 2755 . . . . . . . 8 𝑓 𝐴
5 nfv 1840 . . . . . . . 8 𝑓Fun
64, 5nfim 1822 . . . . . . 7 𝑓(𝐴 → Fun )
7 eleq1 2686 . . . . . . . 8 (𝑓 = → (𝑓𝐴𝐴))
8 funeq 5872 . . . . . . . 8 (𝑓 = → (Fun 𝑓 ↔ Fun ))
97, 8imbi12d 334 . . . . . . 7 (𝑓 = → ((𝑓𝐴 → Fun 𝑓) ↔ (𝐴 → Fun )))
101, 6, 9cbval 2270 . . . . . 6 (∀𝑓(𝑓𝐴 → Fun 𝑓) ↔ ∀(𝐴 → Fun ))
11 df-ral 2912 . . . . . 6 (∀𝑓𝐴 Fun 𝑓 ↔ ∀𝑓(𝑓𝐴 → Fun 𝑓))
12 df-ral 2912 . . . . . 6 (∀𝐴 Fun ↔ ∀(𝐴 → Fun ))
1310, 11, 123bitr4i 292 . . . . 5 (∀𝑓𝐴 Fun 𝑓 ↔ ∀𝐴 Fun )
14 bnj1385.1 . . . . 5 (𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)
15 bnj1385.5 . . . . 5 (𝜑′ ↔ ∀𝐴 Fun )
1613, 14, 153bitr4i 292 . . . 4 (𝜑𝜑′)
17 nfv 1840 . . . . . 6 (𝑓𝐴 → ∀𝑔𝐴 (𝑓𝐷) = (𝑔𝐷))
18 nfv 1840 . . . . . . . 8 𝑓(𝐸) = (𝑔𝐸)
193, 18nfral 2940 . . . . . . 7 𝑓𝑔𝐴 (𝐸) = (𝑔𝐸)
204, 19nfim 1822 . . . . . 6 𝑓(𝐴 → ∀𝑔𝐴 (𝐸) = (𝑔𝐸))
21 dmeq 5289 . . . . . . . . . . . . 13 (𝑓 = → dom 𝑓 = dom )
2221ineq1d 3796 . . . . . . . . . . . 12 (𝑓 = → (dom 𝑓 ∩ dom 𝑔) = (dom ∩ dom 𝑔))
23 bnj1385.2 . . . . . . . . . . . 12 𝐷 = (dom 𝑓 ∩ dom 𝑔)
24 bnj1385.6 . . . . . . . . . . . 12 𝐸 = (dom ∩ dom 𝑔)
2522, 23, 243eqtr4g 2680 . . . . . . . . . . 11 (𝑓 = 𝐷 = 𝐸)
2625reseq2d 5361 . . . . . . . . . 10 (𝑓 = → (𝑓𝐷) = (𝑓𝐸))
27 reseq1 5355 . . . . . . . . . 10 (𝑓 = → (𝑓𝐸) = (𝐸))
2826, 27eqtrd 2655 . . . . . . . . 9 (𝑓 = → (𝑓𝐷) = (𝐸))
2925reseq2d 5361 . . . . . . . . 9 (𝑓 = → (𝑔𝐷) = (𝑔𝐸))
3028, 29eqeq12d 2636 . . . . . . . 8 (𝑓 = → ((𝑓𝐷) = (𝑔𝐷) ↔ (𝐸) = (𝑔𝐸)))
3130ralbidv 2981 . . . . . . 7 (𝑓 = → (∀𝑔𝐴 (𝑓𝐷) = (𝑔𝐷) ↔ ∀𝑔𝐴 (𝐸) = (𝑔𝐸)))
327, 31imbi12d 334 . . . . . 6 (𝑓 = → ((𝑓𝐴 → ∀𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)) ↔ (𝐴 → ∀𝑔𝐴 (𝐸) = (𝑔𝐸))))
3317, 20, 32cbval 2270 . . . . 5 (∀𝑓(𝑓𝐴 → ∀𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)) ↔ ∀(𝐴 → ∀𝑔𝐴 (𝐸) = (𝑔𝐸)))
34 df-ral 2912 . . . . 5 (∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷) ↔ ∀𝑓(𝑓𝐴 → ∀𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))
35 df-ral 2912 . . . . 5 (∀𝐴𝑔𝐴 (𝐸) = (𝑔𝐸) ↔ ∀(𝐴 → ∀𝑔𝐴 (𝐸) = (𝑔𝐸)))
3633, 34, 353bitr4i 292 . . . 4 (∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷) ↔ ∀𝐴𝑔𝐴 (𝐸) = (𝑔𝐸))
3716, 36anbi12i 732 . . 3 ((𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)) ↔ (𝜑′ ∧ ∀𝐴𝑔𝐴 (𝐸) = (𝑔𝐸)))
38 bnj1385.3 . . 3 (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))
39 bnj1385.7 . . 3 (𝜓′ ↔ (𝜑′ ∧ ∀𝐴𝑔𝐴 (𝐸) = (𝑔𝐸)))
4037, 38, 393bitr4i 292 . 2 (𝜓𝜓′)
4115, 24, 39bnj1383 30645 . 2 (𝜓′ → Fun 𝐴)
4240, 41sylbi 207 1 (𝜓 → Fun 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2907  cin 3558   cuni 4407  dom cdm 5079  cres 5081  Fun wfun 5846
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-res 5091  df-iota 5815  df-fun 5854  df-fv 5860
This theorem is referenced by:  bnj1386  30647
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