Theorem functhinclem3 49428

Description: Lemma for functhinc 49430. 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 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
3		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
4	2, 3	oveq12d 7387	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
5	2	fveq2d 6844	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑥) = (𝐹‘𝑋))
6	3	fveq2d 6844	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑦) = (𝐹‘𝑌))
7	5, 6	oveq12d 7387	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
8	4, 7	xpeq12d 5662	. . . 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 7402	. . . . . 6 ⊢ (𝑋𝐻𝑌) ∈ V
12		ovex 7402	. . . . . 6 ⊢ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∈ V
13	11, 12	xpex 7709	. . . . 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 7521	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌) = ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))))
16		eqid 2729	. . . 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 48829	. . 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 7039	1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃*wmo 2531  Vcvv 3444  ∅c0 4292   × cxp 5629  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374

This theorem is referenced by:  functhinclem4  49429