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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1502 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1502.1 | ⊢ (𝜑 → Fun 𝐹) |
bnj1502.2 | ⊢ (𝜑 → 𝐺 ⊆ 𝐹) |
bnj1502.3 | ⊢ (𝜑 → 𝐴 ∈ dom 𝐺) |
Ref | Expression |
---|---|
bnj1502 | ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1502.1 | . 2 ⊢ (𝜑 → Fun 𝐹) | |
2 | bnj1502.2 | . 2 ⊢ (𝜑 → 𝐺 ⊆ 𝐹) | |
3 | bnj1502.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ dom 𝐺) | |
4 | funssfv 6790 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 dom cdm 5589 Fun wfun 6425 ‘cfv 6431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-res 5601 df-iota 6389 df-fun 6433 df-fv 6439 |
This theorem is referenced by: bnj570 32879 bnj929 32910 bnj1450 33024 bnj1501 33041 |
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