Theorem caofcan 37544

Description: Transfer a cancellation law like mulcan 10543 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		ffn 5958	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝑇 → 𝐹 Fn 𝐴)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		caofcan.3	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
5		ffn 5958	. . . . . . 7 ⊢ (𝐺:𝐴⟶𝑆 → 𝐺 Fn 𝐴)
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
7		caofcan.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		inidm 3784	. . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9		eqidd 2611	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
10		eqidd 2611	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝐺‘𝑤))
11	3, 6, 7, 7, 8, 9, 10	ofval 6804	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑤) = ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))
12		caofcan.4	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
13		ffn 5958	. . . . . . 7 ⊢ (𝐻:𝐴⟶𝑆 → 𝐻 Fn 𝐴)
14	12, 13	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn 𝐴)
15		eqidd 2611	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) = (𝐻‘𝑤))
16	3, 14, 7, 7, 8, 9, 15	ofval 6804	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅𝐻)‘𝑤) = ((𝐹‘𝑤)𝑅(𝐻‘𝑤)))
17	11, 16	eqeq12d 2625	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ∘𝑓 𝑅𝐺)‘𝑤) = ((𝐹 ∘𝑓 𝑅𝐻)‘𝑤) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐹‘𝑤)𝑅(𝐻‘𝑤))))
18		simpl 472	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝜑)
19	1	ffvelrnda 6267	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑇)
20	4	ffvelrnda 6267	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
21	12	ffvelrnda 6267	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
22		caofcan.5	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))
23	22	caovcang 6733	. . . . 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)))
24	18, 19, 20, 21, 23	syl13anc 1320	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐹‘𝑤)𝑅(𝐻‘𝑤)) ↔ (𝐺‘𝑤) = (𝐻‘𝑤)))
25	17, 24	bitrd 267	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ∘𝑓 𝑅𝐺)‘𝑤) = ((𝐹 ∘𝑓 𝑅𝐻)‘𝑤) ↔ (𝐺‘𝑤) = (𝐻‘𝑤)))
26	25	ralbidva 2968	. 2 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 ((𝐹 ∘𝑓 𝑅𝐺)‘𝑤) = ((𝐹 ∘𝑓 𝑅𝐻)‘𝑤) ↔ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤) = (𝐻‘𝑤)))
27	3, 6, 7, 7, 8	offn 6806	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) Fn 𝐴)
28	3, 14, 7, 7, 8	offn 6806	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐻) Fn 𝐴)
29		eqfnfv 6219	. . 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)))
30	27, 28, 29	syl2anc 691	. 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) = (𝐹 ∘𝑓 𝑅𝐻) ↔ ∀𝑤 ∈ 𝐴 ((𝐹 ∘𝑓 𝑅𝐺)‘𝑤) = ((𝐹 ∘𝑓 𝑅𝐻)‘𝑤)))
31		eqfnfv 6219	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐻 Fn 𝐴) → (𝐺 = 𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤) = (𝐻‘𝑤)))
32	6, 14, 31	syl2anc 691	. 2 ⊢ (𝜑 → (𝐺 = 𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤) = (𝐻‘𝑤)))
33	26, 30, 32	3bitr4d 299	1 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) = (𝐹 ∘𝑓 𝑅𝐻) ↔ 𝐺 = 𝐻))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795

This theorem is referenced by: (None)