Theorem ofrfval 7409

Description: Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

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

Assertion

Ref	Expression
ofrfval	⊢ (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 ofrfval

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		offval.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		fnex 6972	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
4	1, 2, 3	syl2anc 586	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
5		offval.2	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
6		offval.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		fnex 6972	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊) → 𝐺 ∈ V)
8	5, 6, 7	syl2anc 586	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
9		dmeq 5765	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
10		dmeq 5765	. . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
11	9, 10	ineqan12d 4189	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
12		fveq1 6662	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
13		fveq1 6662	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥))
14	12, 13	breqan12d 5073	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
15	11, 14	raleqbidv 3400	. . . 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)))
16		df-ofr 7402	. . . 4 ⊢ ∘r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)}
17	15, 16	brabga 5412	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥)))
18	4, 8, 17	syl2anc 586	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥)))
19	1	fndmd 6449	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
20	5	fndmd 6449	. . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝐵)
21	19, 20	ineq12d 4188	. . . 4 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
22		offval.5	. . . 4 ⊢ (𝐴 ∩ 𝐵) = 𝑆
23	21, 22	syl6eq 2870	. . 3 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
24	23	raleqdv 3414	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
25		inss1 4203	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
26	22, 25	eqsstrri 4000	. . . . . 6 ⊢ 𝑆 ⊆ 𝐴
27	26	sseli 3961	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐴)
28		offval.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
29	27, 28	sylan2 594	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐹‘𝑥) = 𝐶)
30		inss2 4204	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
31	22, 30	eqsstrri 4000	. . . . . 6 ⊢ 𝑆 ⊆ 𝐵
32	31	sseli 3961	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐵)
33		offval.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)
34	32, 33	sylan2 594	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑥) = 𝐷)
35	29, 34	breq12d 5070	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ 𝐶𝑅𝐷))
36	35	ralbidva 3194	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))
37	18, 24, 36	3bitrd 307	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∀wral 3136  Vcvv 3493   ∩ cin 3933   class class class wbr 5057  dom cdm 5548   Fn wfn 6343  ‘cfv 6348   ∘_r cofr 7400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ofr 7402

This theorem is referenced by:  ofrval  7411  ofrfval2  7419  caofref  7427  caofrss  7434  caoftrn  7436  ofsubge0  11629  pwsle  16757  pwsleval  16758  psrbaglesupp  20140  psrbagcon  20143  psrbaglefi  20144  psrlidm  20175  0plef  24265  0pledm  24266  itg1ge0  24279  mbfi1fseqlem5  24312  xrge0f  24324  itg2ge0  24328  itg2lea  24337  itg2splitlem  24341  itg2monolem1  24343  itg2mono  24346  itg2i1fseqle  24347  itg2i1fseq  24348  itg2addlem  24351  itg2cnlem1  24354  itg2addnclem  34935