Theorem caofcan 39492

Description: Transfer a cancellation law like mulcan 11015 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)

Hypotheses

Ref	Expression
caofcan.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
caofcan.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑇)
caofcan.3	⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
caofcan.4	⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
caofcan.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))

Assertion

Ref	Expression
caofcan	⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) = (𝐹 ∘𝑓 𝑅𝐻) ↔ 𝐺 = 𝐻))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐹   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem caofcan

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofcan.2	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝑇)
2	1	ffnd 6294	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		caofcan.3	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffnd 6294	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
5		caofcan.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		inidm 4043	. . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7		eqidd 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
8		eqidd 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝐺‘𝑤))
9	2, 4, 5, 5, 6, 7, 8	ofval 7185	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑤) = ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))
10		caofcan.4	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
11	10	ffnd 6294	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn 𝐴)
12		eqidd 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) = (𝐻‘𝑤))
13	2, 11, 5, 5, 6, 7, 12	ofval 7185	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅𝐻)‘𝑤) = ((𝐹‘𝑤)𝑅(𝐻‘𝑤)))
14	9, 13	eqeq12d 2793	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ∘𝑓 𝑅𝐺)‘𝑤) = ((𝐹 ∘𝑓 𝑅𝐻)‘𝑤) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐹‘𝑤)𝑅(𝐻‘𝑤))))
15		simpl 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝜑)
16	1	ffvelrnda 6625	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑇)
17	3	ffvelrnda 6625	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
18	10	ffvelrnda 6625	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
19		caofcan.5	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))
20	19	caovcang 7114	. . . . 5 ⊢ ((𝜑 ∧ ((𝐹‘𝑤) ∈ 𝑇 ∧ (𝐺‘𝑤) ∈ 𝑆 ∧ (𝐻‘𝑤) ∈ 𝑆)) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐹‘𝑤)𝑅(𝐻‘𝑤)) ↔ (𝐺‘𝑤) = (𝐻‘𝑤)))
21	15, 16, 17, 18, 20	syl13anc 1440	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐹‘𝑤)𝑅(𝐻‘𝑤)) ↔ (𝐺‘𝑤) = (𝐻‘𝑤)))
22	14, 21	bitrd 271	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ∘𝑓 𝑅𝐺)‘𝑤) = ((𝐹 ∘𝑓 𝑅𝐻)‘𝑤) ↔ (𝐺‘𝑤) = (𝐻‘𝑤)))
23	22	ralbidva 3167	. 2 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 ((𝐹 ∘𝑓 𝑅𝐺)‘𝑤) = ((𝐹 ∘𝑓 𝑅𝐻)‘𝑤) ↔ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤) = (𝐻‘𝑤)))
24	2, 4, 5, 5, 6	offn 7187	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) Fn 𝐴)
25	2, 11, 5, 5, 6	offn 7187	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐻) Fn 𝐴)
26		eqfnfv 6576	. . 3 ⊢ (((𝐹 ∘𝑓 𝑅𝐺) Fn 𝐴 ∧ (𝐹 ∘𝑓 𝑅𝐻) Fn 𝐴) → ((𝐹 ∘𝑓 𝑅𝐺) = (𝐹 ∘𝑓 𝑅𝐻) ↔ ∀𝑤 ∈ 𝐴 ((𝐹 ∘𝑓 𝑅𝐺)‘𝑤) = ((𝐹 ∘𝑓 𝑅𝐻)‘𝑤)))
27	24, 25, 26	syl2anc 579	. 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) = (𝐹 ∘𝑓 𝑅𝐻) ↔ ∀𝑤 ∈ 𝐴 ((𝐹 ∘𝑓 𝑅𝐺)‘𝑤) = ((𝐹 ∘𝑓 𝑅𝐻)‘𝑤)))
28		eqfnfv 6576	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐻 Fn 𝐴) → (𝐺 = 𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤) = (𝐻‘𝑤)))
29	4, 11, 28	syl2anc 579	. 2 ⊢ (𝜑 → (𝐺 = 𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤) = (𝐻‘𝑤)))
30	23, 27, 29	3bitr4d 303	1 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) = (𝐹 ∘𝑓 𝑅𝐻) ↔ 𝐺 = 𝐻))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090   Fn wfn 6132  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924   ∘_𝑓 cof 7174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176

This theorem is referenced by: (None)