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Theorem ofrfvalg 7412
 Description: Value of a relation applied to two functions. Originally part of ofrfval 7414, this version assumes the functions are sets rather than their domains, avoiding ax-rep 5156. (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 5743 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4 dmeq 5743 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
53, 4ineqan12d 4119 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
6 fveq1 6657 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
7 fveq1 6657 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
86, 7breqan12d 5048 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
95, 8raleqbidv 3319 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥) ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
10 df-ofr 7406 . . . 4 r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥)}
119, 10brabga 5391 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐹r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
121, 2, 11syl2anc 587 . 2 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
13 ofrfvalg.1 . . . . . 6 (𝜑𝐹 Fn 𝐴)
1413fndmd 6438 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
15 ofrfvalg.2 . . . . . 6 (𝜑𝐺 Fn 𝐵)
1615fndmd 6438 . . . . 5 (𝜑 → dom 𝐺 = 𝐵)
1714, 16ineq12d 4118 . . . 4 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
18 ofrfvalg.5 . . . 4 (𝐴𝐵) = 𝑆
1917, 18eqtrdi 2809 . . 3 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
2019raleqdv 3329 . 2 (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
21 inss1 4133 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
2218, 21eqsstrri 3927 . . . . . 6 𝑆𝐴
2322sseli 3888 . . . . 5 (𝑥𝑆𝑥𝐴)
24 ofrfvalg.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
2523, 24sylan2 595 . . . 4 ((𝜑𝑥𝑆) → (𝐹𝑥) = 𝐶)
26 inss2 4134 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
2718, 26eqsstrri 3927 . . . . . 6 𝑆𝐵
2827sseli 3888 . . . . 5 (𝑥𝑆𝑥𝐵)
29 ofrfvalg.7 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3028, 29sylan2 595 . . . 4 ((𝜑𝑥𝑆) → (𝐺𝑥) = 𝐷)
3125, 30breq12d 5045 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ 𝐶𝑅𝐷))
3231ralbidva 3125 . 2 (𝜑 → (∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
3312, 20, 323bitrd 308 1 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ∩ cin 3857   class class class wbr 5032  dom cdm 5524   Fn wfn 6330  ‘cfv 6335   ∘r cofr 7404 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 2113  ax-9 2121  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-dm 5534  df-iota 6294  df-fn 6338  df-fv 6343  df-ofr 7406 This theorem is referenced by:  ofrfval  7414  pwsle  16823  pwsleval  16824  psrbaglesupp  20686  psrbaglefi  20694
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