Theorem updjudhcoinlf 9850

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 9849	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
5	4	ffnd 6664	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐴 ⊔ 𝐵))
6		inlresf 9832	. . . 4 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
7		ffn 6663	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) → (inl ↾ 𝐴) Fn 𝐴)
8	6, 7	mp1i 13	. . 3 ⊢ (𝜑 → (inl ↾ 𝐴) Fn 𝐴)
9		frn 6670	. . . 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 6611	. . 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 1374	. 2 ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
13	1	ffnd 6664	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14		fvco2 6932	. . . 4 ⊢ (((inl ↾ 𝐴) Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
15	8, 14	sylan 581	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
16		fvres 6854	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
17	16	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
18	17	fveq2d 6839	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐻‘(inl‘𝑎)))
19		fveqeq2 6844	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → ((1st ‘𝑥) = ∅ ↔ (1st ‘(inl‘𝑎)) = ∅))
20		2fveq3 6840	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → (𝐹‘(2nd ‘𝑥)) = (𝐹‘(2nd ‘(inl‘𝑎))))
21		2fveq3 6840	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → (𝐺‘(2nd ‘𝑥)) = (𝐺‘(2nd ‘(inl‘𝑎))))
22	19, 20, 21	ifbieq12d 4496	. . . . . . 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 9845	. . . . . . . . 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 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 2772	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = (𝐹‘(2nd ‘(inl‘𝑎))))
29		djulcl 9828	. . . . . 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 9846	. . . . . . . 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 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (2nd ‘(inl‘𝑎)) ∈ 𝐴)
36	31, 35	ffvelcdmd 7032	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) ∈ 𝐶)
37	3, 28, 30, 36	fvmptd2 6951	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘(inl‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
38	18, 37	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
39	33	fveq2d 6839	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) = (𝐹‘𝑎))
40	15, 38, 39	3eqtrd 2776	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐹‘𝑎))
41	12, 13, 40	eqfnfvd 6981	1 ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  ∅c0 4274  ifcif 4467   ↦ cmpt 5167  ran crn 5626   ↾ cres 5627   ∘ ccom 5629   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  1^st c1st 7934  2^nd c2nd 7935   ⊔ cdju 9816  inlcinl 9817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-1st 7936  df-2nd 7937  df-1o 8399  df-dju 9819  df-inl 9820

This theorem is referenced by:  updjud  9852