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Mirrors > Home > MPE Home > Th. List > Mathboxes > opidon2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of mndpfo 17922 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
opidon2OLD.1 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
opidon2OLD | ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ dom dom 𝐺 = dom dom 𝐺 | |
2 | 1 | opidonOLD 35011 | . 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺) |
3 | opidon2OLD.1 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
4 | forn 6586 | . . . 4 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → ran 𝐺 = dom dom 𝐺) | |
5 | 3, 4 | syl5req 2866 | . . 3 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → dom dom 𝐺 = 𝑋) |
6 | xpeq12 5573 | . . . . . . 7 ⊢ ((dom dom 𝐺 = 𝑋 ∧ dom dom 𝐺 = 𝑋) → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋)) | |
7 | 6 | anidms 567 | . . . . . 6 ⊢ (dom dom 𝐺 = 𝑋 → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋)) |
8 | foeq2 6580 | . . . . . 6 ⊢ ((dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋) → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺)) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺)) |
10 | foeq3 6581 | . . . . 5 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→𝑋)) | |
11 | 9, 10 | bitrd 280 | . . . 4 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→𝑋)) |
12 | 11 | biimpd 230 | . . 3 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → 𝐺:(𝑋 × 𝑋)–onto→𝑋)) |
13 | 5, 12 | mpcom 38 | . 2 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → 𝐺:(𝑋 × 𝑋)–onto→𝑋) |
14 | 2, 13 | syl 17 | 1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 × cxp 5546 dom cdm 5548 ran crn 5549 –onto→wfo 6346 ExId cexid 35003 Magmacmagm 35007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-ov 7148 df-exid 35004 df-mgmOLD 35008 |
This theorem is referenced by: exidreslem 35036 |
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