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Mirrors > Home > MPE Home > Th. List > Mathboxes > functhinclem3 | Structured version Visualization version GIF version |
Description: Lemma for functhinc 48845. The mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.) |
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))) |
Ref | Expression |
---|---|
functhinclem3 | ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
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 6911 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑥) = (𝐹‘𝑋)) |
6 | 3 | fveq2d 6911 | . . . . . 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 5720 | . . . 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 7772 | . . . . 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 2735 | . . . 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 48678 | . . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃*wmo 2536 Vcvv 3478 ∅c0 4339 × cxp 5687 ⟶wf 6559 ‘cfv 6563 (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 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: functhinclem4 48844 |
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