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Mirrors > Home > MPE Home > Th. List > Mathboxes > functhinclem2 | Structured version Visualization version GIF version |
Description: Lemma for functhinc 45942. (Contributed by Zhi Wang, 1-Oct-2024.) |
Ref | Expression |
---|---|
functhinclem2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
functhinclem2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
functhinclem2.1 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅)) |
Ref | Expression |
---|---|
functhinclem2 | ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | functhinclem2.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | functhinclem2.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
3 | functhinclem2.1 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅)) | |
4 | simpl 486 | . . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
5 | 4 | fveq2d 6699 | . . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐹‘𝑥) = (𝐹‘𝑋)) |
6 | simpr 488 | . . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
7 | 6 | fveq2d 6699 | . . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑌)) |
8 | 5, 7 | oveq12d 7209 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
9 | 8 | eqeq1d 2738 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅)) |
10 | oveq12 7200 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) | |
11 | 10 | eqeq1d 2738 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥𝐻𝑦) = ∅ ↔ (𝑋𝐻𝑌) = ∅)) |
12 | 9, 11 | imbi12d 348 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅) ↔ (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))) |
13 | 12 | rspc2gv 3536 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅) → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))) |
14 | 13 | imp 410 | . 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅)) → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅)) |
15 | 1, 2, 3, 14 | syl21anc 838 | 1 ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∅c0 4223 ‘cfv 6358 (class class class)co 7191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 |
This theorem is referenced by: functhinclem4 45941 functhinc 45942 |
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