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Theorem bnj1385 31354
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 2009 . . . . . . 7 (𝑓𝐴 → Fun 𝑓)
2 bnj1385.4 . . . . . . . . . 10 (𝑥𝐴 → ∀𝑓 𝑥𝐴)
32nfcii 2898 . . . . . . . . 9 𝑓𝐴
43nfcri 2901 . . . . . . . 8 𝑓 𝐴
5 nfv 2009 . . . . . . . 8 𝑓Fun
64, 5nfim 1995 . . . . . . 7 𝑓(𝐴 → Fun )
7 eleq1w 2827 . . . . . . . 8 (𝑓 = → (𝑓𝐴𝐴))
8 funeq 6090 . . . . . . . 8 (𝑓 = → (Fun 𝑓 ↔ Fun ))
97, 8imbi12d 335 . . . . . . 7 (𝑓 = → ((𝑓𝐴 → Fun 𝑓) ↔ (𝐴 → Fun )))
101, 6, 9cbval 2376 . . . . . 6 (∀𝑓(𝑓𝐴 → Fun 𝑓) ↔ ∀(𝐴 → Fun ))
11 df-ral 3060 . . . . . 6 (∀𝑓𝐴 Fun 𝑓 ↔ ∀𝑓(𝑓𝐴 → Fun 𝑓))
12 df-ral 3060 . . . . . 6 (∀𝐴 Fun ↔ ∀(𝐴 → Fun ))
1310, 11, 123bitr4i 294 . . . . 5 (∀𝑓𝐴 Fun 𝑓 ↔ ∀𝐴 Fun )
14 bnj1385.1 . . . . 5 (𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)
15 bnj1385.5 . . . . 5 (𝜑′ ↔ ∀𝐴 Fun )
1613, 14, 153bitr4i 294 . . . 4 (𝜑𝜑′)
17 nfv 2009 . . . . . 6 (𝑓𝐴 → ∀𝑔𝐴 (𝑓𝐷) = (𝑔𝐷))
18 nfv 2009 . . . . . . . 8 𝑓(𝐸) = (𝑔𝐸)
193, 18nfral 3092 . . . . . . 7 𝑓𝑔𝐴 (𝐸) = (𝑔𝐸)
204, 19nfim 1995 . . . . . 6 𝑓(𝐴 → ∀𝑔𝐴 (𝐸) = (𝑔𝐸))
21 dmeq 5494 . . . . . . . . . . . . 13 (𝑓 = → dom 𝑓 = dom )
2221ineq1d 3977 . . . . . . . . . . . 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 2824 . . . . . . . . . . 11 (𝑓 = 𝐷 = 𝐸)
2625reseq2d 5567 . . . . . . . . . 10 (𝑓 = → (𝑓𝐷) = (𝑓𝐸))
27 reseq1 5561 . . . . . . . . . 10 (𝑓 = → (𝑓𝐸) = (𝐸))
2826, 27eqtrd 2799 . . . . . . . . 9 (𝑓 = → (𝑓𝐷) = (𝐸))
2925reseq2d 5567 . . . . . . . . 9 (𝑓 = → (𝑔𝐷) = (𝑔𝐸))
3028, 29eqeq12d 2780 . . . . . . . 8 (𝑓 = → ((𝑓𝐷) = (𝑔𝐷) ↔ (𝐸) = (𝑔𝐸)))
3130ralbidv 3133 . . . . . . 7 (𝑓 = → (∀𝑔𝐴 (𝑓𝐷) = (𝑔𝐷) ↔ ∀𝑔𝐴 (𝐸) = (𝑔𝐸)))
327, 31imbi12d 335 . . . . . 6 (𝑓 = → ((𝑓𝐴 → ∀𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)) ↔ (𝐴 → ∀𝑔𝐴 (𝐸) = (𝑔𝐸))))
3317, 20, 32cbval 2376 . . . . 5 (∀𝑓(𝑓𝐴 → ∀𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)) ↔ ∀(𝐴 → ∀𝑔𝐴 (𝐸) = (𝑔𝐸)))
34 df-ral 3060 . . . . 5 (∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷) ↔ ∀𝑓(𝑓𝐴 → ∀𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))
35 df-ral 3060 . . . . 5 (∀𝐴𝑔𝐴 (𝐸) = (𝑔𝐸) ↔ ∀(𝐴 → ∀𝑔𝐴 (𝐸) = (𝑔𝐸)))
3633, 34, 353bitr4i 294 . . . 4 (∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷) ↔ ∀𝐴𝑔𝐴 (𝐸) = (𝑔𝐸))
3716, 36anbi12i 620 . . 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 294 . 2 (𝜓𝜓′)
4115, 24, 39bnj1383 31353 . 2 (𝜓′ → Fun 𝐴)
4240, 41sylbi 208 1 (𝜓 → Fun 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wcel 2155  wral 3055  cin 3733   cuni 4596  dom cdm 5279  cres 5281  Fun wfun 6064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-res 5291  df-iota 6033  df-fun 6072  df-fv 6078
This theorem is referenced by:  bnj1386  31355
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