updjudhcoinrg - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > updjudhcoinrg		Structured version Visualization version GIF version

Theorem updjudhcoinrg 9928

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 9926	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
5	4	ffnd 6719	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐴 ⊔ 𝐵))
6		inrresf 9911	. . . 4 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵)
7		ffn 6718	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
8	6, 7	mp1i 13	. . 3 ⊢ (𝜑 → (inr ↾ 𝐵) Fn 𝐵)
9		frn 6725	. . . 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 6668	. . 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 1372	. 2 ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) Fn 𝐵)
13	2	ffnd 6719	. 2 ⊢ (𝜑 → 𝐺 Fn 𝐵)
14		fvco2 6989	. . . 4 ⊢ (((inr ↾ 𝐵) Fn 𝐵 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
15	8, 14	sylan 581	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
16		fvres 6911	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
17	16	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
18	17	fveq2d 6896	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐻‘(inr‘𝑏)))
19		fveqeq2 6901	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → ((1^st ‘𝑥) = ∅ ↔ (1^st ‘(inr‘𝑏)) = ∅))
20		2fveq3 6897	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐹‘(2^nd ‘𝑥)) = (𝐹‘(2^nd ‘(inr‘𝑏))))
21		2fveq3 6897	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐺‘(2^nd ‘𝑥)) = (𝐺‘(2^nd ‘(inr‘𝑏))))
22	19, 20, 21	ifbieq12d 4557	. . . . . . 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 483	. . . . . 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 9924	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐵 → (1^st ‘(inr‘𝑏)) = 1_o)
25		1n0 8488	. . . . . . . . . . . 12 ⊢ 1_o ≠ ∅
26	25	neii 2943	. . . . . . . . . . 11 ⊢ ¬ 1_o = ∅
27		eqeq1 2737	. . . . . . . . . . 11 ⊢ ((1^st ‘(inr‘𝑏)) = 1_o → ((1^st ‘(inr‘𝑏)) = ∅ ↔ 1_o = ∅))
28	26, 27	mtbiri 327	. . . . . . . . . 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 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ (1^st ‘(inr‘𝑏)) = ∅)
31	30	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → ¬ (1^st ‘(inr‘𝑏)) = ∅)
32	31	iffalsed 4540	. . . . . 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 2773	. . . . 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 9906	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 → (inr‘𝑏) ∈ (𝐴 ⊔ 𝐵))
35	34	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (inr‘𝑏) ∈ (𝐴 ⊔ 𝐵))
36	2	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐺:𝐵⟶𝐶)
37		2ndinr 9925	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 → (2^nd ‘(inr‘𝑏)) = 𝑏)
38	37	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (2^nd ‘(inr‘𝑏)) = 𝑏)
39		simpr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
40	38, 39	eqeltrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (2^nd ‘(inr‘𝑏)) ∈ 𝐵)
41	36, 40	ffvelcdmd 7088	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐺‘(2^nd ‘(inr‘𝑏))) ∈ 𝐶)
42	3, 33, 35, 41	fvmptd2 7007	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘(inr‘𝑏)) = (𝐺‘(2^nd ‘(inr‘𝑏))))
43	18, 42	eqtrd 2773	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐺‘(2^nd ‘(inr‘𝑏))))
44	38	fveq2d 6896	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐺‘(2^nd ‘(inr‘𝑏))) = (𝐺‘𝑏))
45	15, 43, 44	3eqtrd 2777	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐺‘𝑏))
46	12, 13, 45	eqfnfvd 7036	1 ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3949 ∅c0 4323 ifcif 4529 ↦ cmpt 5232 ran crn 5678 ↾ cres 5679 ∘ ccom 5681 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 1^st c1st 7973 2^nd c2nd 7974 1_oc1o 8459 ⊔ cdju 9893 inrcinr 9895

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-om 7856 df-1st 7975 df-2nd 7976 df-1o 8466 df-dju 9896 df-inr 9898

This theorem is referenced by: updjud 9929

W3C validator