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Theorem fundmge2nop0 13216
Description: A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fundmge2nop 13217 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 15795. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Proof shortened by AV, 15-Nov-2021.)
Assertion
Ref Expression
fundmge2nop0 ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → ¬ 𝐺 ∈ (V × V))

Proof of Theorem fundmge2nop0
Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmexg 7047 . . . . . 6 (𝐺 ∈ V → dom 𝐺 ∈ V)
2 hashge2el2dif 13203 . . . . . . 7 ((dom 𝐺 ∈ V ∧ 2 ≤ (#‘dom 𝐺)) → ∃𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺 𝑎𝑏)
32ex 450 . . . . . 6 (dom 𝐺 ∈ V → (2 ≤ (#‘dom 𝐺) → ∃𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺 𝑎𝑏))
41, 3syl 17 . . . . 5 (𝐺 ∈ V → (2 ≤ (#‘dom 𝐺) → ∃𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺 𝑎𝑏))
5 df-ne 2791 . . . . . . 7 (𝑎𝑏 ↔ ¬ 𝑎 = 𝑏)
6 elvv 5140 . . . . . . . . . . 11 (𝐺 ∈ (V × V) ↔ ∃𝑥𝑦 𝐺 = ⟨𝑥, 𝑦⟩)
7 difeq1 3701 . . . . . . . . . . . . . . . . 17 (𝐺 = ⟨𝑥, 𝑦⟩ → (𝐺 ∖ {∅}) = (⟨𝑥, 𝑦⟩ ∖ {∅}))
87funeqd 5871 . . . . . . . . . . . . . . . 16 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (𝐺 ∖ {∅}) ↔ Fun (⟨𝑥, 𝑦⟩ ∖ {∅})))
9 opwo0id 4923 . . . . . . . . . . . . . . . . . . 19 𝑥, 𝑦⟩ = (⟨𝑥, 𝑦⟩ ∖ {∅})
109eqcomi 2630 . . . . . . . . . . . . . . . . . 18 (⟨𝑥, 𝑦⟩ ∖ {∅}) = ⟨𝑥, 𝑦
1110funeqi 5870 . . . . . . . . . . . . . . . . 17 (Fun (⟨𝑥, 𝑦⟩ ∖ {∅}) ↔ Fun ⟨𝑥, 𝑦⟩)
12 dmeq 5286 . . . . . . . . . . . . . . . . . . . . 21 (𝐺 = ⟨𝑥, 𝑦⟩ → dom 𝐺 = dom ⟨𝑥, 𝑦⟩)
1312eleq2d 2684 . . . . . . . . . . . . . . . . . . . 20 (𝐺 = ⟨𝑥, 𝑦⟩ → (𝑎 ∈ dom 𝐺𝑎 ∈ dom ⟨𝑥, 𝑦⟩))
1412eleq2d 2684 . . . . . . . . . . . . . . . . . . . 20 (𝐺 = ⟨𝑥, 𝑦⟩ → (𝑏 ∈ dom 𝐺𝑏 ∈ dom ⟨𝑥, 𝑦⟩))
1513, 14anbi12d 746 . . . . . . . . . . . . . . . . . . 19 (𝐺 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) ↔ (𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩)))
16 eqid 2621 . . . . . . . . . . . . . . . . . . . . . 22 𝑥, 𝑦⟩ = ⟨𝑥, 𝑦
17 vex 3189 . . . . . . . . . . . . . . . . . . . . . 22 𝑥 ∈ V
18 vex 3189 . . . . . . . . . . . . . . . . . . . . . 22 𝑦 ∈ V
1916, 17, 18funopdmsn 6372 . . . . . . . . . . . . . . . . . . . . 21 ((Fun ⟨𝑥, 𝑦⟩ ∧ 𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩) → 𝑎 = 𝑏)
20193expb 1263 . . . . . . . . . . . . . . . . . . . 20 ((Fun ⟨𝑥, 𝑦⟩ ∧ (𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩)) → 𝑎 = 𝑏)
2120expcom 451 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩) → (Fun ⟨𝑥, 𝑦⟩ → 𝑎 = 𝑏))
2215, 21syl6bi 243 . . . . . . . . . . . . . . . . . 18 (𝐺 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (Fun ⟨𝑥, 𝑦⟩ → 𝑎 = 𝑏)))
2322com23 86 . . . . . . . . . . . . . . . . 17 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → 𝑎 = 𝑏)))
2411, 23syl5bi 232 . . . . . . . . . . . . . . . 16 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (⟨𝑥, 𝑦⟩ ∖ {∅}) → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → 𝑎 = 𝑏)))
258, 24sylbid 230 . . . . . . . . . . . . . . 15 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (𝐺 ∖ {∅}) → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → 𝑎 = 𝑏)))
2625com23 86 . . . . . . . . . . . . . 14 (𝐺 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (Fun (𝐺 ∖ {∅}) → 𝑎 = 𝑏)))
2726impd 447 . . . . . . . . . . . . 13 (𝐺 = ⟨𝑥, 𝑦⟩ → (((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → 𝑎 = 𝑏))
2827exlimivv 1857 . . . . . . . . . . . 12 (∃𝑥𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → (((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → 𝑎 = 𝑏))
2928com12 32 . . . . . . . . . . 11 (((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → (∃𝑥𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → 𝑎 = 𝑏))
306, 29syl5bi 232 . . . . . . . . . 10 (((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → (𝐺 ∈ (V × V) → 𝑎 = 𝑏))
3130con3d 148 . . . . . . . . 9 (((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → (¬ 𝑎 = 𝑏 → ¬ 𝐺 ∈ (V × V)))
3231ex 450 . . . . . . . 8 ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (Fun (𝐺 ∖ {∅}) → (¬ 𝑎 = 𝑏 → ¬ 𝐺 ∈ (V × V))))
3332com23 86 . . . . . . 7 ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (¬ 𝑎 = 𝑏 → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V))))
345, 33syl5bi 232 . . . . . 6 ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (𝑎𝑏 → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V))))
3534rexlimivv 3029 . . . . 5 (∃𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺 𝑎𝑏 → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V)))
364, 35syl6 35 . . . 4 (𝐺 ∈ V → (2 ≤ (#‘dom 𝐺) → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V))))
3736com13 88 . . 3 (Fun (𝐺 ∖ {∅}) → (2 ≤ (#‘dom 𝐺) → (𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))))
3837imp 445 . 2 ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (𝐺 ∈ V → ¬ 𝐺 ∈ (V × V)))
39 prcnel 3204 . 2 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
4038, 39pm2.61d1 171 1 ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → ¬ 𝐺 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  Vcvv 3186  cdif 3553  c0 3893  {csn 4150  cop 4156   class class class wbr 4615   × cxp 5074  dom cdm 5076  Fun wfun 5843  cfv 5849  cle 10022  2c2 11017  #chash 13060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-card 8712  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-n0 11240  df-xnn0 11311  df-z 11325  df-uz 11635  df-fz 12272  df-hash 13061
This theorem is referenced by:  fundmge2nop  13217  fun2dmnop0  13218  funvtxdmge2val  25798  funiedgdmge2val  25799  funvtxdmge2valOLD  25806  funiedgdmge2valOLD  25807
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