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Mirrors > Home > MPE Home > Th. List > 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.) (Revised by AV, 23-Mar-2024.) |
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 6673 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
3 | ssidd 3968 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴) | |
4 | 1 | fdmd 6680 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
5 | 3, 4 | sseqtrrd 3986 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) |
6 | funfvima2 7182 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑋 ∈ 𝐴 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴))) | |
7 | 2, 5, 6 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝐴 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴))) |
8 | 7 | imp 408 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3911 dom cdm 5634 “ cima 5637 Fun wfun 6491 ⟶wf 6493 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 |
This theorem is referenced by: imo72b2lem1 42530 fundcmpsurbijinjpreimafv 45685 |
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