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Mirrors > Home > MPE Home > Th. List > fundif | Structured version Visualization version GIF version |
Description: A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.) |
Ref | Expression |
---|---|
fundif | ⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldif 5828 | . . 3 ⊢ (Rel 𝐹 → Rel (𝐹 ∖ 𝐴)) | |
2 | brdif 5201 | . . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐴)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦)) | |
3 | brdif 5201 | . . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐴)𝑧 ↔ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧)) | |
4 | pm2.27 42 | . . . . . . . 8 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧)) | |
5 | 4 | ad2ant2r 747 | . . . . . . 7 ⊢ (((𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦) ∧ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧)) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧)) |
6 | 2, 3, 5 | syl2anb 598 | . . . . . 6 ⊢ ((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧)) |
7 | 6 | com12 32 | . . . . 5 ⊢ (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧)) |
8 | 7 | alimi 1808 | . . . 4 ⊢ (∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑧((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧)) |
9 | 8 | 2alimi 1809 | . . 3 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧)) |
10 | 1, 9 | anim12i 613 | . 2 ⊢ ((Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)) → (Rel (𝐹 ∖ 𝐴) ∧ ∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧))) |
11 | dffun2 6573 | . 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))) | |
12 | dffun2 6573 | . 2 ⊢ (Fun (𝐹 ∖ 𝐴) ↔ (Rel (𝐹 ∖ 𝐴) ∧ ∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧))) | |
13 | 10, 11, 12 | 3imtr4i 292 | 1 ⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1535 ∖ cdif 3960 class class class wbr 5148 Rel wrel 5694 Fun wfun 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-fun 6565 |
This theorem is referenced by: fundmge2nop 14539 fun2dmnop 14541 |
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