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Mirrors > Home > MPE Home > Th. List > fun2dmnop0 | Structured version Visualization version GIF version |
Description: A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 14488 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 17119. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.) |
Ref | Expression |
---|---|
fun2dmnop.a | ⊢ 𝐴 ∈ V |
fun2dmnop.b | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
fun2dmnop0 | ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1188 | . . . 4 ⊢ (((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) ∧ 𝐺 ∈ V) → Fun (𝐺 ∖ {∅})) | |
2 | dmexg 7907 | . . . . . 6 ⊢ (𝐺 ∈ V → dom 𝐺 ∈ V) | |
3 | 2 | adantl 480 | . . . . 5 ⊢ (((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) ∧ 𝐺 ∈ V) → dom 𝐺 ∈ V) |
4 | fun2dmnop.a | . . . . . . . . 9 ⊢ 𝐴 ∈ V | |
5 | fun2dmnop.b | . . . . . . . . 9 ⊢ 𝐵 ∈ V | |
6 | 4, 5 | prss 4819 | . . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) ↔ {𝐴, 𝐵} ⊆ dom 𝐺) |
7 | simpl 481 | . . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 ∈ dom 𝐺) | |
8 | 6, 7 | sylbir 234 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ⊆ dom 𝐺 → 𝐴 ∈ dom 𝐺) |
9 | 8 | 3ad2ant3 1132 | . . . . . 6 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → 𝐴 ∈ dom 𝐺) |
10 | 9 | adantr 479 | . . . . 5 ⊢ (((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) ∧ 𝐺 ∈ V) → 𝐴 ∈ dom 𝐺) |
11 | simpr 483 | . . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐵 ∈ dom 𝐺) | |
12 | 6, 11 | sylbir 234 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ⊆ dom 𝐺 → 𝐵 ∈ dom 𝐺) |
13 | 12 | 3ad2ant3 1132 | . . . . . 6 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → 𝐵 ∈ dom 𝐺) |
14 | 13 | adantr 479 | . . . . 5 ⊢ (((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) ∧ 𝐺 ∈ V) → 𝐵 ∈ dom 𝐺) |
15 | simpl2 1189 | . . . . 5 ⊢ (((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) ∧ 𝐺 ∈ V) → 𝐴 ≠ 𝐵) | |
16 | 3, 10, 14, 15 | nehash2 14467 | . . . 4 ⊢ (((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) ∧ 𝐺 ∈ V) → 2 ≤ (♯‘dom 𝐺)) |
17 | fundmge2nop0 14485 | . . . 4 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → ¬ 𝐺 ∈ (V × V)) | |
18 | 1, 16, 17 | syl2anc 582 | . . 3 ⊢ (((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) ∧ 𝐺 ∈ V) → ¬ 𝐺 ∈ (V × V)) |
19 | 18 | ex 411 | . 2 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))) |
20 | prcnel 3487 | . 2 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V)) | |
21 | 19, 20 | pm2.61d1 180 | 1 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2930 Vcvv 3463 ∖ cdif 3936 ⊆ wss 3939 ∅c0 4318 {csn 4624 {cpr 4626 class class class wbr 5143 × cxp 5670 dom cdm 5672 Fun wfun 6537 ‘cfv 6543 ≤ cle 11279 2c2 12297 ♯chash 14321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13517 df-hash 14322 |
This theorem is referenced by: fun2dmnop 14488 funvtxdm2val 28870 funiedgdm2val 28871 |
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