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Theorem caofid0r 7641
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 6666 . 2 (𝜑𝐹 Fn 𝐴)
4 caofid0.3 . . 3 (𝜑𝐵𝑊)
5 fnconstg 6727 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
64, 5syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
7 eqidd 2737 . 2 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
8 fvconst2g 7147 . . 3 ((𝐵𝑊𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
94, 8sylan 580 . 2 ((𝜑𝑤𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
10 caofid0r.5 . . . 4 ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝑥)
1110ralrimiva 3141 . . 3 (𝜑 → ∀𝑥𝑆 (𝑥𝑅𝐵) = 𝑥)
122ffvelcdmda 7031 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
13 oveq1 7358 . . . . 5 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝐵) = ((𝐹𝑤)𝑅𝐵))
14 id 22 . . . . 5 (𝑥 = (𝐹𝑤) → 𝑥 = (𝐹𝑤))
1513, 14eqeq12d 2752 . . . 4 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝐵) = 𝑥 ↔ ((𝐹𝑤)𝑅𝐵) = (𝐹𝑤)))
1615rspccva 3578 . . 3 ((∀𝑥𝑆 (𝑥𝑅𝐵) = 𝑥 ∧ (𝐹𝑤) ∈ 𝑆) → ((𝐹𝑤)𝑅𝐵) = (𝐹𝑤))
1711, 12, 16syl2an2r 683 . 2 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅𝐵) = (𝐹𝑤))
181, 3, 6, 3, 7, 9, 17offveq 7633 1 (𝜑 → (𝐹f 𝑅(𝐴 × {𝐵})) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3062  {csn 4584   × cxp 5629   Fn wfn 6488  wf 6489  cfv 6493  (class class class)co 7351  f cof 7607
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-of 7609
This theorem is referenced by:  psrlidm  21318  mndvrid  21689  lfl1sc  37478
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