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Theorem updjudhcoinrg 9924

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 9922	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
5	4	ffnd 6715	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐴 ⊔ 𝐵))
6		inrresf 9907	. . . 4 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵)
7		ffn 6714	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
8	6, 7	mp1i 13	. . 3 ⊢ (𝜑 → (inr ↾ 𝐵) Fn 𝐵)
9		frn 6721	. . . 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 6664	. . 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 6715	. 2 ⊢ (𝜑 → 𝐺 Fn 𝐵)
14		fvco2 6985	. . . 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 6907	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
17	16	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
18	17	fveq2d 6892	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐻‘(inr‘𝑏)))
19		fveqeq2 6897	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → ((1^st ‘𝑥) = ∅ ↔ (1^st ‘(inr‘𝑏)) = ∅))
20		2fveq3 6893	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐹‘(2^nd ‘𝑥)) = (𝐹‘(2^nd ‘(inr‘𝑏))))
21		2fveq3 6893	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐺‘(2^nd ‘𝑥)) = (𝐺‘(2^nd ‘(inr‘𝑏))))
22	19, 20, 21	ifbieq12d 4555	. . . . . . 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 9920	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐵 → (1^st ‘(inr‘𝑏)) = 1_o)
25		1n0 8484	. . . . . . . . . . . 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 4538	. . . . . 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 9902	. . . . . 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 9921	. . . . . . . 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 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐺‘(2^nd ‘(inr‘𝑏))) ∈ 𝐶)
42	3, 33, 35, 41	fvmptd2 7003	. . . 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 6892	. . 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 7032	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 3947 ∅c0 4321 ifcif 4527 ↦ cmpt 5230 ran crn 5676 ↾ cres 5677 ∘ ccom 5679 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 1^st c1st 7969 2^nd c2nd 7970 1_oc1o 8455 ⊔ cdju 9889 inrcinr 9891

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 5298 ax-nul 5305 ax-pr 5426 ax-un 7721

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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7852 df-1st 7971 df-2nd 7972 df-1o 8462 df-dju 9892 df-inr 9894

This theorem is referenced by: updjud 9925

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