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Mirrors > Home > MPE Home > Th. List > dffo3 | Structured version Visualization version GIF version |
Description: An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.) |
Ref | Expression |
---|---|
dffo3 | ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffo2 6569 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) | |
2 | ffn 6487 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | fnrnfv 6700 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) | |
4 | 3 | eqeq1d 2800 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵)) |
6 | dfbi2 478 | . . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))) | |
7 | simpr 488 | . . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = (𝐹‘𝑥)) | |
8 | ffvelrn 6826 | . . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) | |
9 | 8 | adantr 484 | . . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ 𝐵) |
10 | 7, 9 | eqeltrd 2890 | . . . . . . . . 9 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 ∈ 𝐵) |
11 | 10 | rexlimdva2 3246 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵)) |
12 | 11 | biantrurd 536 | . . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → ((𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) ↔ ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))))) |
13 | 6, 12 | bitr4id 293 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))) |
14 | 13 | albidv 1921 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))) |
15 | abeq1 2923 | . . . . 5 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵 ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵)) | |
16 | df-ral 3111 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) | |
17 | 14, 15, 16 | 3bitr4g 317 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) |
18 | 5, 17 | bitrd 282 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (ran 𝐹 = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) |
19 | 18 | pm5.32i 578 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) |
20 | 1, 19 | bitri 278 | 1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∈ wcel 2111 {cab 2776 ∀wral 3106 ∃wrex 3107 ran crn 5520 Fn wfn 6319 ⟶wf 6320 –onto→wfo 6322 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 |
This theorem is referenced by: dffo4 6846 foelrn 6849 foco2 6850 fcofo 7022 foov 7302 resixpfo 8483 fofinf1o 8783 wdom2d 9028 brwdom3 9030 isf32lem9 9772 hsmexlem2 9838 cnref1o 12372 wwlktovfo 14313 1arith 16253 fullestrcsetc 17393 fullsetcestrc 17408 orbsta 18435 symgextfo 18542 symgfixfo 18559 pwssplit1 19824 znf1o 20243 cygznlem3 20261 scmatfo 21135 m2cpmfo 21361 pm2mpfo 21419 recosf1o 25127 efif1olem4 25137 dvdsmulf1o 25779 wlkswwlksf1o 27665 wwlksnextsurj 27686 clwlkclwwlkfo 27794 clwwlkfo 27835 eucrctshift 28028 frgrncvvdeqlem9 28092 numclwwlk1lem2fo 28143 subfacp1lem3 32542 cvmfolem 32639 finixpnum 35042 wessf1ornlem 41811 projf1o 41825 sumnnodd 42272 dvnprodlem1 42588 fourierdlem54 42802 nnfoctbdjlem 43094 isomenndlem 43169 sprsymrelfo 44014 prproropf1o 44024 isomuspgrlem2d 44349 uspgrsprfo 44376 1arymaptfo 45057 2arymaptfo 45068 rrx2xpref1o 45132 |
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