Theorem updjudhcoinlf 9691

Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022.)

Hypotheses

Ref	Expression
updjud.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
updjud.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐶)
updjudhf.h	⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))

Assertion

Ref	Expression
updjudhcoinlf	⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝑥,𝐹

Allowed substitution hints:   𝐺(𝑥)   𝐻(𝑥)

Proof of Theorem updjudhcoinlf

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		updjud.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
2		updjud.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)
3		updjudhf.h	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))
4	1, 2, 3	updjudhf 9690	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
5	4	ffnd 6599	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐴 ⊔ 𝐵))
6		inlresf 9673	. . . 4 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
7		ffn 6598	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) → (inl ↾ 𝐴) Fn 𝐴)
8	6, 7	mp1i 13	. . 3 ⊢ (𝜑 → (inl ↾ 𝐴) Fn 𝐴)
9		frn 6605	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) → ran (inl ↾ 𝐴) ⊆ (𝐴 ⊔ 𝐵))
10	6, 9	mp1i 13	. . 3 ⊢ (𝜑 → ran (inl ↾ 𝐴) ⊆ (𝐴 ⊔ 𝐵))
11		fnco 6547	. . 3 ⊢ ((𝐻 Fn (𝐴 ⊔ 𝐵) ∧ (inl ↾ 𝐴) Fn 𝐴 ∧ ran (inl ↾ 𝐴) ⊆ (𝐴 ⊔ 𝐵)) → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
12	5, 8, 10, 11	syl3anc 1370	. 2 ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
13	1	ffnd 6599	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14		fvco2 6862	. . . 4 ⊢ (((inl ↾ 𝐴) Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
15	8, 14	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
16		fvres 6790	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
17	16	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
18	17	fveq2d 6775	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐻‘(inl‘𝑎)))
19		fveqeq2 6780	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → ((1st ‘𝑥) = ∅ ↔ (1st ‘(inl‘𝑎)) = ∅))
20		2fveq3 6776	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → (𝐹‘(2nd ‘𝑥)) = (𝐹‘(2nd ‘(inl‘𝑎))))
21		2fveq3 6776	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → (𝐺‘(2nd ‘𝑥)) = (𝐺‘(2nd ‘(inl‘𝑎))))
22	19, 20, 21	ifbieq12d 4493	. . . . . . 7 ⊢ (𝑥 = (inl‘𝑎) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))))
23	22	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))))
24		1stinl 9686	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → (1st ‘(inl‘𝑎)) = ∅)
25	24	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1st ‘(inl‘𝑎)) = ∅)
26	25	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → (1st ‘(inl‘𝑎)) = ∅)
27	26	iftrued 4473	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))) = (𝐹‘(2nd ‘(inl‘𝑎))))
28	23, 27	eqtrd 2780	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = (𝐹‘(2nd ‘(inl‘𝑎))))
29		djulcl 9669	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → (inl‘𝑎) ∈ (𝐴 ⊔ 𝐵))
30	29	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (inl‘𝑎) ∈ (𝐴 ⊔ 𝐵))
31	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐹:𝐴⟶𝐶)
32		2ndinl 9687	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐴 → (2nd ‘(inl‘𝑎)) = 𝑎)
33	32	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (2nd ‘(inl‘𝑎)) = 𝑎)
34		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
35	33, 34	eqeltrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (2nd ‘(inl‘𝑎)) ∈ 𝐴)
36	31, 35	ffvelrnd 6959	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) ∈ 𝐶)
37	3, 28, 30, 36	fvmptd2 6880	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘(inl‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
38	18, 37	eqtrd 2780	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
39	33	fveq2d 6775	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) = (𝐹‘𝑎))
40	15, 38, 39	3eqtrd 2784	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐹‘𝑎))
41	12, 13, 40	eqfnfvd 6909	1 ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110   ⊆ wss 3892  ∅c0 4262  ifcif 4465   ↦ cmpt 5162  ran crn 5591   ↾ cres 5592   ∘ ccom 5594   Fn wfn 6427  ⟶wf 6428  ‘cfv 6432  1^st c1st 7822  2^nd c2nd 7823   ⊔ cdju 9657  inlcinl 9658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-1st 7824  df-2nd 7825  df-1o 8288  df-dju 9660  df-inl 9661

This theorem is referenced by:  updjud  9693