Theorem updjudhcoinlf 9828

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 9827	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
5	4	ffnd 6653	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐴 ⊔ 𝐵))
6		inlresf 9810	. . . 4 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
7		ffn 6652	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) → (inl ↾ 𝐴) Fn 𝐴)
8	6, 7	mp1i 13	. . 3 ⊢ (𝜑 → (inl ↾ 𝐴) Fn 𝐴)
9		frn 6659	. . . 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 6600	. . 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 1373	. 2 ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
13	1	ffnd 6653	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14		fvco2 6920	. . . 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 6841	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
17	16	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
18	17	fveq2d 6826	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐻‘(inl‘𝑎)))
19		fveqeq2 6831	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → ((1st ‘𝑥) = ∅ ↔ (1st ‘(inl‘𝑎)) = ∅))
20		2fveq3 6827	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → (𝐹‘(2nd ‘𝑥)) = (𝐹‘(2nd ‘(inl‘𝑎))))
21		2fveq3 6827	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → (𝐺‘(2nd ‘𝑥)) = (𝐺‘(2nd ‘(inl‘𝑎))))
22	19, 20, 21	ifbieq12d 4505	. . . . . . 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 481	. . . . . 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 9823	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → (1st ‘(inl‘𝑎)) = ∅)
25	24	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1st ‘(inl‘𝑎)) = ∅)
26	25	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → (1st ‘(inl‘𝑎)) = ∅)
27	26	iftrued 4484	. . . . . 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 2764	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = (𝐹‘(2nd ‘(inl‘𝑎))))
29		djulcl 9806	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → (inl‘𝑎) ∈ (𝐴 ⊔ 𝐵))
30	29	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (inl‘𝑎) ∈ (𝐴 ⊔ 𝐵))
31	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐹:𝐴⟶𝐶)
32		2ndinl 9824	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐴 → (2nd ‘(inl‘𝑎)) = 𝑎)
33	32	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (2nd ‘(inl‘𝑎)) = 𝑎)
34		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
35	33, 34	eqeltrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (2nd ‘(inl‘𝑎)) ∈ 𝐴)
36	31, 35	ffvelcdmd 7019	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) ∈ 𝐶)
37	3, 28, 30, 36	fvmptd2 6938	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘(inl‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
38	18, 37	eqtrd 2764	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
39	33	fveq2d 6826	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) = (𝐹‘𝑎))
40	15, 38, 39	3eqtrd 2768	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐹‘𝑎))
41	12, 13, 40	eqfnfvd 6968	1 ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3903  ∅c0 4284  ifcif 4476   ↦ cmpt 5173  ran crn 5620   ↾ cres 5621   ∘ ccom 5623   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  1^st c1st 7922  2^nd c2nd 7923   ⊔ cdju 9794  inlcinl 9795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-1st 7924  df-2nd 7925  df-1o 8388  df-dju 9797  df-inl 9798

This theorem is referenced by:  updjud  9830