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Mirrors > Home > MPE Home > Th. List > f1ocpbl | Structured version Visualization version GIF version |
Description: An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
f1ocpbl.f | ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋) |
Ref | Expression |
---|---|
f1ocpbl | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocpbl.f | . . 3 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋) | |
2 | 1 | f1ocpbllem 16800 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
3 | oveq12 7168 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 + 𝐵) = (𝐶 + 𝐷)) | |
4 | 3 | fveq2d 6677 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))) |
5 | 2, 4 | syl6bi 255 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7159 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-f1o 6365 df-fv 6366 df-ov 7162 |
This theorem is referenced by: xpsadd 16850 xpsmul 16851 imasmndf1 17953 imasgrpf1 18219 imasgim 39706 |
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