![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > f1ovscpbl | Structured version Visualization version GIF version |
Description: An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.) |
Ref | Expression |
---|---|
f1ocpbl.f | ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋) |
Ref | Expression |
---|---|
f1ovscpbl | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐶) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocpbl.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋) | |
2 | f1of1 6848 | . . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝑋 → 𝐹:𝑉–1-1→𝑋) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝑉–1-1→𝑋) |
4 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐹:𝑉–1-1→𝑋) |
5 | simpr2 1194 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉) | |
6 | simpr3 1195 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
7 | f1fveq 7282 | . . 3 ⊢ ((𝐹:𝑉–1-1→𝑋 ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶)) | |
8 | 4, 5, 6, 7 | syl12anc 837 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶)) |
9 | oveq2 7439 | . . 3 ⊢ (𝐵 = 𝐶 → (𝐴 + 𝐵) = (𝐴 + 𝐶)) | |
10 | 9 | fveq2d 6911 | . 2 ⊢ (𝐵 = 𝐶 → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶))) |
11 | 8, 10 | biimtrdi 253 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐶) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 –1-1→wf1 6560 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 |
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-pr 5438 |
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-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-f1o 6570 df-fv 6571 df-ov 7434 |
This theorem is referenced by: xpsvsca 17624 |
Copyright terms: Public domain | W3C validator |