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Theorem updjudhcoinrg 9930

Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second 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((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))

**Assertion**
Ref	Expression
updjudhcoinrg	⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝜑,𝑥 𝑥,𝐹 𝑥,𝐺 Allowed substitution hint: 𝐻(𝑥)

Proof of Theorem updjudhcoinrg Dummy variable 𝑏 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((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))))
4	1, 2, 3	updjudhf 9928	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
5	4	ffnd 6718	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐴 ⊔ 𝐵))
6		inrresf 9913	. . . 4 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵)
7		ffn 6717	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
8	6, 7	mp1i 13	. . 3 ⊢ (𝜑 → (inr ↾ 𝐵) Fn 𝐵)
9		frn 6724	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) → ran (inr ↾ 𝐵) ⊆ (𝐴 ⊔ 𝐵))
10	6, 9	mp1i 13	. . 3 ⊢ (𝜑 → ran (inr ↾ 𝐵) ⊆ (𝐴 ⊔ 𝐵))
11		fnco 6667	. . 3 ⊢ ((𝐻 Fn (𝐴 ⊔ 𝐵) ∧ (inr ↾ 𝐵) Fn 𝐵 ∧ ran (inr ↾ 𝐵) ⊆ (𝐴 ⊔ 𝐵)) → (𝐻 ∘ (inr ↾ 𝐵)) Fn 𝐵)
12	5, 8, 10, 11	syl3anc 1371	. 2 ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) Fn 𝐵)
13	2	ffnd 6718	. 2 ⊢ (𝜑 → 𝐺 Fn 𝐵)
14		fvco2 6988	. . . 4 ⊢ (((inr ↾ 𝐵) Fn 𝐵 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
15	8, 14	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
16		fvres 6910	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
17	16	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
18	17	fveq2d 6895	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐻‘(inr‘𝑏)))
19		fveqeq2 6900	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → ((1^st ‘𝑥) = ∅ ↔ (1^st ‘(inr‘𝑏)) = ∅))
20		2fveq3 6896	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐹‘(2^nd ‘𝑥)) = (𝐹‘(2^nd ‘(inr‘𝑏))))
21		2fveq3 6896	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐺‘(2^nd ‘𝑥)) = (𝐺‘(2^nd ‘(inr‘𝑏))))
22	19, 20, 21	ifbieq12d 4556	. . . . . . 7 ⊢ (𝑥 = (inr‘𝑏) → if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))) = if((1^st ‘(inr‘𝑏)) = ∅, (𝐹‘(2^nd ‘(inr‘𝑏))), (𝐺‘(2^nd ‘(inr‘𝑏)))))
23	22	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))) = if((1^st ‘(inr‘𝑏)) = ∅, (𝐹‘(2^nd ‘(inr‘𝑏))), (𝐺‘(2^nd ‘(inr‘𝑏)))))
24		1stinr 9926	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐵 → (1^st ‘(inr‘𝑏)) = 1_o)
25		1n0 8490	. . . . . . . . . . . 12 ⊢ 1_o ≠ ∅
26	25	neii 2942	. . . . . . . . . . 11 ⊢ ¬ 1_o = ∅
27		eqeq1 2736	. . . . . . . . . . 11 ⊢ ((1^st ‘(inr‘𝑏)) = 1_o → ((1^st ‘(inr‘𝑏)) = ∅ ↔ 1_o = ∅))
28	26, 27	mtbiri 326	. . . . . . . . . 10 ⊢ ((1^st ‘(inr‘𝑏)) = 1_o → ¬ (1^st ‘(inr‘𝑏)) = ∅)
29	24, 28	syl 17	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝐵 → ¬ (1^st ‘(inr‘𝑏)) = ∅)
30	29	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ (1^st ‘(inr‘𝑏)) = ∅)
31	30	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → ¬ (1^st ‘(inr‘𝑏)) = ∅)
32	31	iffalsed 4539	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1^st ‘(inr‘𝑏)) = ∅, (𝐹‘(2^nd ‘(inr‘𝑏))), (𝐺‘(2^nd ‘(inr‘𝑏)))) = (𝐺‘(2^nd ‘(inr‘𝑏))))
33	23, 32	eqtrd 2772	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1^st ‘𝑥) = ∅, (𝐹‘(2^nd ‘𝑥)), (𝐺‘(2^nd ‘𝑥))) = (𝐺‘(2^nd ‘(inr‘𝑏))))
34		djurcl 9908	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 → (inr‘𝑏) ∈ (𝐴 ⊔ 𝐵))
35	34	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (inr‘𝑏) ∈ (𝐴 ⊔ 𝐵))
36	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐺:𝐵⟶𝐶)
37		2ndinr 9927	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 → (2^nd ‘(inr‘𝑏)) = 𝑏)
38	37	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (2^nd ‘(inr‘𝑏)) = 𝑏)
39		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
40	38, 39	eqeltrd 2833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (2^nd ‘(inr‘𝑏)) ∈ 𝐵)
41	36, 40	ffvelcdmd 7087	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐺‘(2^nd ‘(inr‘𝑏))) ∈ 𝐶)
42	3, 33, 35, 41	fvmptd2 7006	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘(inr‘𝑏)) = (𝐺‘(2^nd ‘(inr‘𝑏))))
43	18, 42	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐺‘(2^nd ‘(inr‘𝑏))))
44	38	fveq2d 6895	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐺‘(2^nd ‘(inr‘𝑏))) = (𝐺‘𝑏))
45	15, 43, 44	3eqtrd 2776	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐺‘𝑏))
46	12, 13, 45	eqfnfvd 7035	1 ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3948 ∅c0 4322 ifcif 4528 ↦ cmpt 5231 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 1^st c1st 7975 2^nd c2nd 7976 1_oc1o 8461 ⊔ cdju 9895 inrcinr 9897

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7858 df-1st 7977 df-2nd 7978 df-1o 8468 df-dju 9898 df-inr 9900

This theorem is referenced by: updjud 9931

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