Theorem updjudhcoinlf 9355

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 9354	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
5	4	ffnd 6510	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐴 ⊔ 𝐵))
6		inlresf 9337	. . . 4 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
7		ffn 6509	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) → (inl ↾ 𝐴) Fn 𝐴)
8	6, 7	mp1i 13	. . 3 ⊢ (𝜑 → (inl ↾ 𝐴) Fn 𝐴)
9		frn 6515	. . . 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 6460	. . 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 1367	. 2 ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
13	1	ffnd 6510	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14		fvco2 6753	. . . 4 ⊢ (((inl ↾ 𝐴) Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
15	8, 14	sylan 582	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
16		fvres 6684	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
17	16	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
18	17	fveq2d 6669	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐻‘(inl‘𝑎)))
19		fveqeq2 6674	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → ((1st ‘𝑥) = ∅ ↔ (1st ‘(inl‘𝑎)) = ∅))
20		2fveq3 6670	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → (𝐹‘(2nd ‘𝑥)) = (𝐹‘(2nd ‘(inl‘𝑎))))
21		2fveq3 6670	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → (𝐺‘(2nd ‘𝑥)) = (𝐺‘(2nd ‘(inl‘𝑎))))
22	19, 20, 21	ifbieq12d 4494	. . . . . . 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 484	. . . . . 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 9350	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → (1st ‘(inl‘𝑎)) = ∅)
25	24	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1st ‘(inl‘𝑎)) = ∅)
26	25	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → (1st ‘(inl‘𝑎)) = ∅)
27	26	iftrued 4475	. . . . . 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 2856	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = (𝐹‘(2nd ‘(inl‘𝑎))))
29		djulcl 9333	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → (inl‘𝑎) ∈ (𝐴 ⊔ 𝐵))
30	29	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (inl‘𝑎) ∈ (𝐴 ⊔ 𝐵))
31	1	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐹:𝐴⟶𝐶)
32		2ndinl 9351	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐴 → (2nd ‘(inl‘𝑎)) = 𝑎)
33	32	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (2nd ‘(inl‘𝑎)) = 𝑎)
34		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
35	33, 34	eqeltrd 2913	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (2nd ‘(inl‘𝑎)) ∈ 𝐴)
36	31, 35	ffvelrnd 6847	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) ∈ 𝐶)
37	3, 28, 30, 36	fvmptd2 6771	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘(inl‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
38	18, 37	eqtrd 2856	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
39	33	fveq2d 6669	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) = (𝐹‘𝑎))
40	15, 38, 39	3eqtrd 2860	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐹‘𝑎))
41	12, 13, 40	eqfnfvd 6800	1 ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ⊆ wss 3936  ∅c0 4291  ifcif 4467   ↦ cmpt 5139  ran crn 5551   ↾ cres 5552   ∘ ccom 5554   Fn wfn 6345  ⟶wf 6346  ‘cfv 6350  1^st c1st 7681  2^nd c2nd 7682   ⊔ cdju 9321  inlcinl 9322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-1st 7683  df-2nd 7684  df-1o 8096  df-dju 9324  df-inl 9325

This theorem is referenced by:  updjud  9357