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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 6119 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) | |
5 | 1, 2, 3, 4 | syl3anc 1318 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 dom cdm 5038 Fun wfun 5798 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: bnj570 30229 bnj929 30260 bnj1450 30372 bnj1501 30389 |
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