Theorem updjudhcoinrg 9923

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((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘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((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))
4	1, 2, 3	updjudhf 9921	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
5	4	ffnd 6714	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐴 ⊔ 𝐵))
6		inrresf 9906	. . . 4 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵)
7		ffn 6713	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
8	6, 7	mp1i 13	. . 3 ⊢ (𝜑 → (inr ↾ 𝐵) Fn 𝐵)
9		frn 6720	. . . 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 6663	. . 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 6714	. 2 ⊢ (𝜑 → 𝐺 Fn 𝐵)
14		fvco2 6983	. . . 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 6906	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
17	16	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
18	17	fveq2d 6891	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐻‘(inr‘𝑏)))
19		fveqeq2 6896	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → ((1st ‘𝑥) = ∅ ↔ (1st ‘(inr‘𝑏)) = ∅))
20		2fveq3 6892	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐹‘(2nd ‘𝑥)) = (𝐹‘(2nd ‘(inr‘𝑏))))
21		2fveq3 6892	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐺‘(2nd ‘𝑥)) = (𝐺‘(2nd ‘(inr‘𝑏))))
22	19, 20, 21	ifbieq12d 4554	. . . . . . 7 ⊢ (𝑥 = (inr‘𝑏) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))))
23	22	adantl 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))))
24		1stinr 9919	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐵 → (1st ‘(inr‘𝑏)) = 1o)
25		1n0 8482	. . . . . . . . . . . 12 ⊢ 1o ≠ ∅
26	25	neii 2943	. . . . . . . . . . 11 ⊢ ¬ 1o = ∅
27		eqeq1 2737	. . . . . . . . . . 11 ⊢ ((1st ‘(inr‘𝑏)) = 1o → ((1st ‘(inr‘𝑏)) = ∅ ↔ 1o = ∅))
28	26, 27	mtbiri 327	. . . . . . . . . 10 ⊢ ((1st ‘(inr‘𝑏)) = 1o → ¬ (1st ‘(inr‘𝑏)) = ∅)
29	24, 28	syl 17	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝐵 → ¬ (1st ‘(inr‘𝑏)) = ∅)
30	29	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ (1st ‘(inr‘𝑏)) = ∅)
31	30	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → ¬ (1st ‘(inr‘𝑏)) = ∅)
32	31	iffalsed 4537	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))) = (𝐺‘(2nd ‘(inr‘𝑏))))
33	23, 32	eqtrd 2773	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = (𝐺‘(2nd ‘(inr‘𝑏))))
34		djurcl 9901	. . . . . 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 9920	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 → (2nd ‘(inr‘𝑏)) = 𝑏)
38	37	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (2nd ‘(inr‘𝑏)) = 𝑏)
39		simpr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
40	38, 39	eqeltrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (2nd ‘(inr‘𝑏)) ∈ 𝐵)
41	36, 40	ffvelcdmd 7082	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐺‘(2nd ‘(inr‘𝑏))) ∈ 𝐶)
42	3, 33, 35, 41	fvmptd2 7001	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘(inr‘𝑏)) = (𝐺‘(2nd ‘(inr‘𝑏))))
43	18, 42	eqtrd 2773	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐺‘(2nd ‘(inr‘𝑏))))
44	38	fveq2d 6891	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐺‘(2nd ‘(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 7030	1 ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ⊆ wss 3946  ∅c0 4320  ifcif 4526   ↦ cmpt 5229  ran crn 5675   ↾ cres 5676   ∘ ccom 5678   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  1^st c1st 7967  2^nd c2nd 7968  1_oc1o 8453   ⊔ cdju 9888  inrcinr 9890

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 5297  ax-nul 5304  ax-pr 5425  ax-un 7719

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 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-om 7850  df-1st 7969  df-2nd 7970  df-1o 8460  df-dju 9891  df-inr 9893

This theorem is referenced by:  updjud  9924