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Mathbox for Jonathan Ben-Naim |
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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 6914 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) | |
5 | 1, 2, 3, 4 | syl3anc 1368 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3946 dom cdm 5674 Fun wfun 6540 ‘cfv 6546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-res 5686 df-iota 6498 df-fun 6548 df-fv 6554 |
This theorem is referenced by: bnj570 34763 bnj929 34794 bnj1450 34908 bnj1501 34925 |
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