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Theorem caofid0r 7422
Description: Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
caofref.1 (𝜑𝐴𝑉)
caofref.2 (𝜑𝐹:𝐴𝑆)
caofid0.3 (𝜑𝐵𝑊)
caofid0r.5 ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝑥)
Assertion
Ref Expression
caofid0r (𝜑 → (𝐹f 𝑅(𝐴 × {𝐵})) = 𝐹)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem caofid0r
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofref.1 . 2 (𝜑𝐴𝑉)
2 caofref.2 . . 3 (𝜑𝐹:𝐴𝑆)
32ffnd 6492 . 2 (𝜑𝐹 Fn 𝐴)
4 caofid0.3 . . 3 (𝜑𝐵𝑊)
5 fnconstg 6545 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
64, 5syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
7 eqidd 2802 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
8 fvconst2g 6945 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
94, 8sylan 583 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
10 caofid0r.5 . . . 4 ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝑥)
1110ralrimiva 3152 . . 3 (𝜑 → ∀𝑥𝑆 (𝑥𝑅𝐵) = 𝑥)
122ffvelrnda 6832 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
13 oveq1 7146 . . . . 5 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝐵) = ((𝐹𝑤)𝑅𝐵))
14 id 22 . . . . 5 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
1513, 14eqeq12d 2817 . . . 4 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝐵) = 𝑥 ↔ ((𝐹𝑤)𝑅𝐵) = (𝐹𝑤)))
1615rspccva 3573 . . 3 ((∀𝑥𝑆 (𝑥𝑅𝐵) = 𝑥 ∧ (𝐹𝑤) ∈ 𝑆) → ((𝐹𝑤)𝑅𝐵) = (𝐹𝑤))
1711, 12, 16syl2an2r 684 . 2 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅𝐵) = (𝐹𝑤))
181, 3, 6, 3, 7, 9, 17offveq 7414 1 (𝜑 → (𝐹f 𝑅(𝐴 × {𝐵})) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  {csn 4528   × cxp 5521   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  f cof 7391
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393
This theorem is referenced by:  psrlidm  20645  mndvrid  21005  lfl1sc  36379
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