Theorem functhinclem3 47750

Description: Lemma for functhinc 47752. 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 767	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
3		simprr 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
4	2, 3	oveq12d 7429	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
5	2	fveq2d 6894	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑥) = (𝐹‘𝑋))
6	3	fveq2d 6894	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑦) = (𝐹‘𝑌))
7	5, 6	oveq12d 7429	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
8	4, 7	xpeq12d 5706	. . . 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 7444	. . . . . 6 ⊢ (𝑋𝐻𝑌) ∈ V
12		ovex 7444	. . . . . 6 ⊢ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∈ V
13	11, 12	xpex 7742	. . . . 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 7562	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌) = ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))))
16		eqid 2730	. . . 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 47601	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↔ (𝑋𝐺𝑌) = ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))))
20	15, 19	mpbird 256	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
21		functhinclem3.m	. 2 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
22	20, 21	ffvelcdmd 7086	1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∃*wmo 2530  Vcvv 3472  ∅c0 4321   × cxp 5673  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411   ∈ cmpo 7413

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  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-id 5573  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-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416

This theorem is referenced by:  functhinclem4  47751