Theorem caofcan 45266

Description: Transfer a cancellation law like mulcan 11934 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 6702	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		caofcan.3	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffnd 6702	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
5		caofcan.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		inidm 4172	. . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7		eqidd 2762	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
8		eqidd 2762	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝐺‘𝑤))
9	2, 4, 5, 5, 6, 7, 8	ofval 7693	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑤) = ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))
10		caofcan.4	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
11	10	ffnd 6702	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn 𝐴)
12		eqidd 2762	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) = (𝐻‘𝑤))
13	2, 11, 5, 5, 6, 7, 12	ofval 7693	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐻)‘𝑤) = ((𝐹‘𝑤)𝑅(𝐻‘𝑤)))
14	9, 13	eqeq12d 2777	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ∘f 𝑅𝐺)‘𝑤) = ((𝐹 ∘f 𝑅𝐻)‘𝑤) ↔ ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐹‘𝑤)𝑅(𝐻‘𝑤))))
15		simpl 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝜑)
16	1	ffvelcdmda 7076	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑇)
17	3	ffvelcdmda 7076	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
18	10	ffvelcdmda 7076	. . . . 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 7614	. . . . 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 1399	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐹‘𝑤)𝑅(𝐻‘𝑤)) ↔ (𝐺‘𝑤) = (𝐻‘𝑤)))
22	14, 21	bitrd 282	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ∘f 𝑅𝐺)‘𝑤) = ((𝐹 ∘f 𝑅𝐻)‘𝑤) ↔ (𝐺‘𝑤) = (𝐻‘𝑤)))
23	22	ralbidva 3184	. 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 7695	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝐴)
25	2, 11, 5, 5, 6	offn 7695	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐻) Fn 𝐴)
26		eqfnfv 7021	. . 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 596	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f 𝑅𝐻) ↔ ∀𝑤 ∈ 𝐴 ((𝐹 ∘f 𝑅𝐺)‘𝑤) = ((𝐹 ∘f 𝑅𝐻)‘𝑤)))
28		eqfnfv 7021	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐻 Fn 𝐴) → (𝐺 = 𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤) = (𝐻‘𝑤)))
29	4, 11, 28	syl2anc 596	. 2 ⊢ (𝜑 → (𝐺 = 𝐻 ↔ ∀𝑤 ∈ 𝐴 (𝐺‘𝑤) = (𝐻‘𝑤)))
30	23, 27, 29	3bitr4d 314	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f 𝑅𝐻) ↔ 𝐺 = 𝐻))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ∘_f cof 7680

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682

This theorem is used by: (None)