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Mirrors > Home > MPE Home > Th. List > f1otrgds | Structured version Visualization version GIF version |
Description: Convenient lemma for f1otrg 26665. (Contributed by Thierry Arnoux, 19-Mar-2019.) |
Ref | Expression |
---|---|
f1otrkg.p | ⊢ 𝑃 = (Base‘𝐺) |
f1otrkg.d | ⊢ 𝐷 = (dist‘𝐺) |
f1otrkg.i | ⊢ 𝐼 = (Itv‘𝐺) |
f1otrkg.b | ⊢ 𝐵 = (Base‘𝐻) |
f1otrkg.e | ⊢ 𝐸 = (dist‘𝐻) |
f1otrkg.j | ⊢ 𝐽 = (Itv‘𝐻) |
f1otrkg.f | ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝑃) |
f1otrkg.1 | ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓))) |
f1otrkg.2 | ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓)))) |
f1otrgitv.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
f1otrgitv.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
f1otrgds | ⊢ (𝜑 → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1otrkg.1 | . . 3 ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓))) | |
2 | 1 | ralrimivva 3156 | . 2 ⊢ (𝜑 → ∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓))) |
3 | f1otrgitv.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
4 | f1otrgitv.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
5 | oveq1 7142 | . . . . 5 ⊢ (𝑒 = 𝑋 → (𝑒𝐸𝑓) = (𝑋𝐸𝑓)) | |
6 | fveq2 6645 | . . . . . 6 ⊢ (𝑒 = 𝑋 → (𝐹‘𝑒) = (𝐹‘𝑋)) | |
7 | 6 | oveq1d 7150 | . . . . 5 ⊢ (𝑒 = 𝑋 → ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) = ((𝐹‘𝑋)𝐷(𝐹‘𝑓))) |
8 | 5, 7 | eqeq12d 2814 | . . . 4 ⊢ (𝑒 = 𝑋 → ((𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) ↔ (𝑋𝐸𝑓) = ((𝐹‘𝑋)𝐷(𝐹‘𝑓)))) |
9 | oveq2 7143 | . . . . 5 ⊢ (𝑓 = 𝑌 → (𝑋𝐸𝑓) = (𝑋𝐸𝑌)) | |
10 | fveq2 6645 | . . . . . 6 ⊢ (𝑓 = 𝑌 → (𝐹‘𝑓) = (𝐹‘𝑌)) | |
11 | 10 | oveq2d 7151 | . . . . 5 ⊢ (𝑓 = 𝑌 → ((𝐹‘𝑋)𝐷(𝐹‘𝑓)) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌))) |
12 | 9, 11 | eqeq12d 2814 | . . . 4 ⊢ (𝑓 = 𝑌 → ((𝑋𝐸𝑓) = ((𝐹‘𝑋)𝐷(𝐹‘𝑓)) ↔ (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌)))) |
13 | 8, 12 | rspc2v 3581 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌)))) |
14 | 3, 4, 13 | syl2anc 587 | . 2 ⊢ (𝜑 → (∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌)))) |
15 | 2, 14 | mpd 15 | 1 ⊢ (𝜑 → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 Itvcitv 26230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 |
This theorem is referenced by: f1otrg 26665 |
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