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Theorem ofrfvalg 7680
Description: Value of a relation applied to two functions. Originally part of ofrfval 7682, this version assumes the functions are sets rather than their domains, avoiding ax-rep 5284. (Contributed by SN, 5-Aug-2024.)
Hypotheses
Ref Expression
ofrfvalg.1 (𝜑𝐹 Fn 𝐴)
ofrfvalg.2 (𝜑𝐺 Fn 𝐵)
ofrfvalg.3 (𝜑𝐹𝑉)
ofrfvalg.4 (𝜑𝐺𝑊)
ofrfvalg.5 (𝐴𝐵) = 𝑆
ofrfvalg.6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
ofrfvalg.7 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
Assertion
Ref Expression
ofrfvalg (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝑆   𝑥,𝑅
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ofrfvalg
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofrfvalg.3 . . 3 (𝜑𝐹𝑉)
2 ofrfvalg.4 . . 3 (𝜑𝐺𝑊)
3 dmeq 5902 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4 dmeq 5902 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
53, 4ineqan12d 4213 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
6 fveq1 6889 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
7 fveq1 6889 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
86, 7breqan12d 5163 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
95, 8raleqbidv 3340 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥) ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
10 df-ofr 7673 . . . 4 r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥)}
119, 10brabga 5533 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐹r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
121, 2, 11syl2anc 582 . 2 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
13 ofrfvalg.1 . . . . . 6 (𝜑𝐹 Fn 𝐴)
1413fndmd 6653 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
15 ofrfvalg.2 . . . . . 6 (𝜑𝐺 Fn 𝐵)
1615fndmd 6653 . . . . 5 (𝜑 → dom 𝐺 = 𝐵)
1714, 16ineq12d 4212 . . . 4 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
18 ofrfvalg.5 . . . 4 (𝐴𝐵) = 𝑆
1917, 18eqtrdi 2786 . . 3 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
2019raleqdv 3323 . 2 (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
21 inss1 4227 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
2218, 21eqsstrri 4016 . . . . . 6 𝑆𝐴
2322sseli 3977 . . . . 5 (𝑥𝑆𝑥𝐴)
24 ofrfvalg.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
2523, 24sylan2 591 . . . 4 ((𝜑𝑥𝑆) → (𝐹𝑥) = 𝐶)
26 inss2 4228 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
2718, 26eqsstrri 4016 . . . . . 6 𝑆𝐵
2827sseli 3977 . . . . 5 (𝑥𝑆𝑥𝐵)
29 ofrfvalg.7 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3028, 29sylan2 591 . . . 4 ((𝜑𝑥𝑆) → (𝐺𝑥) = 𝐷)
3125, 30breq12d 5160 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ 𝐶𝑅𝐷))
3231ralbidva 3173 . 2 (𝜑 → (∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
3312, 20, 323bitrd 304 1 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wral 3059  cin 3946   class class class wbr 5147  dom cdm 5675   Fn wfn 6537  cfv 6542  r cofr 7671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-dm 5685  df-iota 6494  df-fn 6545  df-fv 6550  df-ofr 7673
This theorem is referenced by:  ofrfval  7682  pwsle  17442  pwsleval  17443  psrbaglesupp  21696  psrbaglefi  21704
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