Theorem functhinclem3 49095

Description: Lemma for functhinc 49097. The mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.)

Hypotheses

Ref	Expression
functhinclem3.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
functhinclem3.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
functhinclem3.m	⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
functhinclem3.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))))
functhinclem3.1	⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))
functhinclem3.2	⊢ (𝜑 → ∃*𝑛 𝑛 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Assertion

Ref	Expression
functhinclem3	⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Distinct variable groups:   𝑛,𝐹   𝑥,𝐹,𝑦   𝑥,𝐻,𝑦   𝑛,𝐽   𝑥,𝐽,𝑦   𝑛,𝑋   𝑥,𝑋,𝑦   𝑛,𝑌   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑛)   𝐵(𝑥,𝑦,𝑛)   𝐺(𝑥,𝑦,𝑛)   𝐻(𝑛)   𝑀(𝑥,𝑦,𝑛)

Proof of Theorem functhinclem3

Step	Hyp	Ref	Expression
1		functhinclem3.g	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))))
2		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
3		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
4	2, 3	oveq12d 7449	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
5	2	fveq2d 6910	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑥) = (𝐹‘𝑋))
6	3	fveq2d 6910	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑦) = (𝐹‘𝑌))
7	5, 6	oveq12d 7449	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
8	4, 7	xpeq12d 5716	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))) = ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))))
9		functhinclem3.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		functhinclem3.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
11		ovex 7464	. . . . . 6 ⊢ (𝑋𝐻𝑌) ∈ V
12		ovex 7464	. . . . . 6 ⊢ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∈ V
13	11, 12	xpex 7773	. . . . 5 ⊢ ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) ∈ V)
15	1, 8, 9, 10, 14	ovmpod 7585	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌) = ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))))
16		eqid 2737	. . . 4 ⊢ ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) = ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
17		functhinclem3.1	. . . 4 ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))
18		functhinclem3.2	. . . 4 ⊢ (𝜑 → ∃*𝑛 𝑛 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
19	16, 17, 18	mofeu 48757	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↔ (𝑋𝐺𝑌) = ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))))
20	15, 19	mpbird 257	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
21		functhinclem3.m	. 2 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
22	20, 21	ffvelcdmd 7105	1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃*wmo 2538  Vcvv 3480  ∅c0 4333   × cxp 5683  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436

This theorem is referenced by:  functhinclem4  49096