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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > funfvima2d | Structured version Visualization version GIF version |
Description: A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
funfvima2d.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
funfvima2d | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfvima2d.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | ffund 6297 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
3 | ssidd 3843 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴) | |
4 | 1 | fdmd 6302 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
5 | 3, 4 | sseqtr4d 3861 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) |
6 | funfvima2 6767 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))) | |
7 | 2, 5, 6 | syl2anc 579 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))) |
8 | 7 | imp 397 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 ⊆ wss 3792 dom cdm 5357 “ cima 5360 Fun wfun 6131 ⟶wf 6133 ‘cfv 6137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-fv 6145 |
This theorem is referenced by: imo72b2lem1 39441 |
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