Theorem ofrfvalg 7618

Description: Value of a relation applied to two functions. Originally part of ofrfval 7620, this version assumes the functions are sets rather than their domains, avoiding ax-rep 5215. (Contributed by SN, 5-Aug-2024.)

Hypotheses

Ref	Expression
ofrfvalg.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
ofrfvalg.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
ofrfvalg.3	⊢ (𝜑 → 𝐹 ∈ 𝑉)
ofrfvalg.4	⊢ (𝜑 → 𝐺 ∈ 𝑊)
ofrfvalg.5	⊢ (𝐴 ∩ 𝐵) = 𝑆
ofrfvalg.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
ofrfvalg.7	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)

Assertion

Ref	Expression
ofrfvalg	⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 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.

Step	Hyp	Ref	Expression
1		ofrfvalg.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		ofrfvalg.4	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
3		dmeq 5842	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4		dmeq 5842	. . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
5	3, 4	ineqan12d 4169	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
6		fveq1 6821	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
7		fveq1 6821	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥))
8	6, 7	breqan12d 5105	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
9	5, 8	raleqbidv 3312	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥)))
10		df-ofr 7611	. . . 4 ⊢ ∘r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)}
11	9, 10	brabga 5472	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥)))
12	1, 2, 11	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥)))
13		ofrfvalg.1	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14	13	fndmd 6586	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
15		ofrfvalg.2	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐵)
16	15	fndmd 6586	. . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝐵)
17	14, 16	ineq12d 4168	. . . 4 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
18		ofrfvalg.5	. . . 4 ⊢ (𝐴 ∩ 𝐵) = 𝑆
19	17, 18	eqtrdi 2782	. . 3 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
20	19	raleqdv 3292	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
21		inss1 4184	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
22	18, 21	eqsstrri 3977	. . . . . 6 ⊢ 𝑆 ⊆ 𝐴
23	22	sseli 3925	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐴)
24		ofrfvalg.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
25	23, 24	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐹‘𝑥) = 𝐶)
26		inss2 4185	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
27	18, 26	eqsstrri 3977	. . . . . 6 ⊢ 𝑆 ⊆ 𝐵
28	27	sseli 3925	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐵)
29		ofrfvalg.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)
30	28, 29	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑥) = 𝐷)
31	25, 30	breq12d 5102	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ 𝐶𝑅𝐷))
32	31	ralbidva 3153	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))
33	12, 20, 32	3bitrd 305	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∩ cin 3896   class class class wbr 5089  dom cdm 5614   Fn wfn 6476  ‘cfv 6481   ∘_r cofr 7609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-dm 5624  df-iota 6437  df-fn 6484  df-fv 6489  df-ofr 7611

This theorem is referenced by:  ofrfval  7620  pwsle  17396  pwsleval  17397  psrbaglesupp  21859  psrbaglefi  21863