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Mirrors > Home > MPE Home > Th. List > mapex | Structured version Visualization version GIF version |
Description: The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.) (Proof shortened by AV, 16-Jun-2025.) |
Ref | Expression |
---|---|
mapex | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)} = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)} | |
2 | 1 | fabexg 7959 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)} ∈ V) |
3 | id 22 | . . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓:𝐴⟶𝐵) | |
4 | 3 | ancli 548 | . . . 4 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)) |
5 | 4 | ss2abi 4077 | . . 3 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)} |
6 | 5 | a1i 11 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)}) |
7 | 2, 6 | ssexd 5330 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 {cab 2712 Vcvv 3478 ⊆ wss 3963 ⟶wf 6559 |
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-pow 5371 ax-pr 5438 ax-un 7754 |
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-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: fnmap 8872 mapvalg 8875 isghmOLD 19247 wksfval 29642 measbase 34178 measval 34179 ismeas 34180 isrnmeas 34181 sticksstones4 42131 sticksstones14 42142 sticksstones20 42148 cnfex 44966 opabresexd 47237 upwlksfval 47979 |
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