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Mirrors > Home > MPE Home > Th. List > dffun9 | Structured version Visualization version GIF version |
Description: Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
dffun9 | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun7 6529 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦)) | |
2 | vex 3448 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
3 | vex 3448 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | brelrn 5898 | . . . . . . 7 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴) |
5 | 4 | pm4.71ri 562 | . . . . . 6 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦)) |
6 | 5 | mobii 2543 | . . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦)) |
7 | df-rmo 3352 | . . . . 5 ⊢ (∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦)) | |
8 | 6, 7 | bitr4i 278 | . . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦) |
9 | 8 | ralbii 3093 | . . 3 ⊢ (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦) |
10 | 9 | anbi2i 624 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)) |
11 | 1, 10 | bitri 275 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∃*wmo 2533 ∀wral 3061 ∃*wrmo 3351 class class class wbr 5106 dom cdm 5634 ran crn 5635 Rel wrel 5639 Fun wfun 6491 |
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-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-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rmo 3352 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-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-fun 6499 |
This theorem is referenced by: brdom4 10471 |
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