Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fundmge2nop0 Structured version   Visualization version   GIF version

Theorem fundmge2nop0 13129
 Description: A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fundmge2nop 13130 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see isstruct 15705. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.)
Assertion
Ref Expression
fundmge2nop0 ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → ¬ 𝐺 ∈ (V × V))

Proof of Theorem fundmge2nop0
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmexg 6989 . . . . . 6 (𝐺 ∈ V → dom 𝐺 ∈ V)
2 hashge2el2dif 13117 . . . . . . 7 ((dom 𝐺 ∈ V ∧ 2 ≤ (#‘dom 𝐺)) → ∃𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺 𝑎𝑏)
32ex 449 . . . . . 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 2782 . . . . . . . 8 (𝑎𝑏 ↔ ¬ 𝑎 = 𝑏)
6 elvv 5100 . . . . . . . . . . . 12 (𝐺 ∈ (V × V) ↔ ∃𝑥𝑦 𝐺 = ⟨𝑥, 𝑦⟩)
7 difeq1 3683 . . . . . . . . . . . . . . . . . 18 (𝐺 = ⟨𝑥, 𝑦⟩ → (𝐺 ∖ {∅}) = (⟨𝑥, 𝑦⟩ ∖ {∅}))
87funeqd 5825 . . . . . . . . . . . . . . . . 17 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (𝐺 ∖ {∅}) ↔ Fun (⟨𝑥, 𝑦⟩ ∖ {∅})))
9 0nelop 4886 . . . . . . . . . . . . . . . . . . . . 21 ¬ ∅ ∈ ⟨𝑥, 𝑦
10 disjsn 4192 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝑥, 𝑦⟩ ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ⟨𝑥, 𝑦⟩)
119, 10mpbir 220 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑥, 𝑦⟩ ∩ {∅}) = ∅
12 disjdif2 3999 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑥, 𝑦⟩ ∩ {∅}) = ∅ → (⟨𝑥, 𝑦⟩ ∖ {∅}) = ⟨𝑥, 𝑦⟩)
1311, 12ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (⟨𝑥, 𝑦⟩ ∖ {∅}) = ⟨𝑥, 𝑦
1413funeqi 5824 . . . . . . . . . . . . . . . . . 18 (Fun (⟨𝑥, 𝑦⟩ ∖ {∅}) ↔ Fun ⟨𝑥, 𝑦⟩)
15 vex 3176 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
16 vex 3176 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
1715, 16funop 6320 . . . . . . . . . . . . . . . . . . 19 (Fun ⟨𝑥, 𝑦⟩ ↔ ∃𝑐(𝑥 = {𝑐} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑐, 𝑐⟩}))
18 eqeq2 2621 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑥, 𝑦⟩ = {⟨𝑐, 𝑐⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ ↔ 𝐺 = {⟨𝑐, 𝑐⟩}))
19 dmeq 5246 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐺 = {⟨𝑐, 𝑐⟩} → dom 𝐺 = dom {⟨𝑐, 𝑐⟩})
20 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐 ∈ V
2120dmsnop 5527 . . . . . . . . . . . . . . . . . . . . . . . . 25 dom {⟨𝑐, 𝑐⟩} = {𝑐}
2219, 21syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐺 = {⟨𝑐, 𝑐⟩} → dom 𝐺 = {𝑐})
23 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (dom 𝐺 = {𝑐} → (𝑎 ∈ dom 𝐺𝑎 ∈ {𝑐}))
24 velsn 4141 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 ∈ {𝑐} ↔ 𝑎 = 𝑐)
2523, 24syl6bb 275 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝐺 = {𝑐} → (𝑎 ∈ dom 𝐺𝑎 = 𝑐))
26 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (dom 𝐺 = {𝑐} → (𝑏 ∈ dom 𝐺𝑏 ∈ {𝑐}))
27 velsn 4141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 ∈ {𝑐} ↔ 𝑏 = 𝑐)
2826, 27syl6bb 275 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (dom 𝐺 = {𝑐} → (𝑏 ∈ dom 𝐺𝑏 = 𝑐))
29 equtr2 1941 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑎 = 𝑐𝑏 = 𝑐) → 𝑎 = 𝑏)
3029a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 = 𝑐𝑏 = 𝑐) → (𝐺 ∈ V → 𝑎 = 𝑏))
3130expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑏 = 𝑐 → (𝑎 = 𝑐 → (𝐺 ∈ V → 𝑎 = 𝑏)))
3228, 31syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (dom 𝐺 = {𝑐} → (𝑏 ∈ dom 𝐺 → (𝑎 = 𝑐 → (𝐺 ∈ V → 𝑎 = 𝑏))))
3332com23 84 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝐺 = {𝑐} → (𝑎 = 𝑐 → (𝑏 ∈ dom 𝐺 → (𝐺 ∈ V → 𝑎 = 𝑏))))
3425, 33sylbid 229 . . . . . . . . . . . . . . . . . . . . . . . . 25 (dom 𝐺 = {𝑐} → (𝑎 ∈ dom 𝐺 → (𝑏 ∈ dom 𝐺 → (𝐺 ∈ V → 𝑎 = 𝑏))))
3534impd 446 . . . . . . . . . . . . . . . . . . . . . . . 24 (dom 𝐺 = {𝑐} → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (𝐺 ∈ V → 𝑎 = 𝑏)))
3622, 35syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 = {⟨𝑐, 𝑐⟩} → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (𝐺 ∈ V → 𝑎 = 𝑏)))
3718, 36syl6bi 242 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑥, 𝑦⟩ = {⟨𝑐, 𝑐⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (𝐺 ∈ V → 𝑎 = 𝑏))))
3837adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = {𝑐} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑐, 𝑐⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (𝐺 ∈ V → 𝑎 = 𝑏))))
3938exlimiv 1845 . . . . . . . . . . . . . . . . . . . 20 (∃𝑐(𝑥 = {𝑐} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑐, 𝑐⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (𝐺 ∈ V → 𝑎 = 𝑏))))
4039com12 32 . . . . . . . . . . . . . . . . . . 19 (𝐺 = ⟨𝑥, 𝑦⟩ → (∃𝑐(𝑥 = {𝑐} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑐, 𝑐⟩}) → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (𝐺 ∈ V → 𝑎 = 𝑏))))
4117, 40syl5bi 231 . . . . . . . . . . . . . . . . . 18 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (𝐺 ∈ V → 𝑎 = 𝑏))))
4214, 41syl5bi 231 . . . . . . . . . . . . . . . . 17 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (⟨𝑥, 𝑦⟩ ∖ {∅}) → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (𝐺 ∈ V → 𝑎 = 𝑏))))
438, 42sylbid 229 . . . . . . . . . . . . . . . 16 (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (𝐺 ∖ {∅}) → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (𝐺 ∈ V → 𝑎 = 𝑏))))
4443com23 84 . . . . . . . . . . . . . . 15 (𝐺 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (Fun (𝐺 ∖ {∅}) → (𝐺 ∈ V → 𝑎 = 𝑏))))
45443impd 1273 . . . . . . . . . . . . . 14 (𝐺 = ⟨𝑥, 𝑦⟩ → (((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅}) ∧ 𝐺 ∈ V) → 𝑎 = 𝑏))
4645exlimivv 1847 . . . . . . . . . . . . 13 (∃𝑥𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → (((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅}) ∧ 𝐺 ∈ V) → 𝑎 = 𝑏))
4746com12 32 . . . . . . . . . . . 12 (((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅}) ∧ 𝐺 ∈ V) → (∃𝑥𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → 𝑎 = 𝑏))
486, 47syl5bi 231 . . . . . . . . . . 11 (((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅}) ∧ 𝐺 ∈ V) → (𝐺 ∈ (V × V) → 𝑎 = 𝑏))
4948con3d 147 . . . . . . . . . 10 (((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅}) ∧ 𝐺 ∈ V) → (¬ 𝑎 = 𝑏 → ¬ 𝐺 ∈ (V × V)))
50493exp 1256 . . . . . . . . 9 ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (Fun (𝐺 ∖ {∅}) → (𝐺 ∈ V → (¬ 𝑎 = 𝑏 → ¬ 𝐺 ∈ (V × V)))))
5150com24 93 . . . . . . . 8 ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (¬ 𝑎 = 𝑏 → (𝐺 ∈ V → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V)))))
525, 51syl5bi 231 . . . . . . 7 ((𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺) → (𝑎𝑏 → (𝐺 ∈ V → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V)))))
5352rexlimivv 3018 . . . . . 6 (∃𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺 𝑎𝑏 → (𝐺 ∈ V → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V))))
5453com12 32 . . . . 5 (𝐺 ∈ V → (∃𝑎 ∈ dom 𝐺𝑏 ∈ dom 𝐺 𝑎𝑏 → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V))))
554, 54syld 46 . . . 4 (𝐺 ∈ V → (2 ≤ (#‘dom 𝐺) → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V))))
5655com13 86 . . 3 (Fun (𝐺 ∖ {∅}) → (2 ≤ (#‘dom 𝐺) → (𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))))
5756imp 444 . 2 ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (𝐺 ∈ V → ¬ 𝐺 ∈ (V × V)))
58 prcnel 3191 . 2 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
5957, 58pm2.61d1 170 1 ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → ¬ 𝐺 ∈ (V × V))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038  Fun wfun 5798  ‘cfv 5804   ≤ cle 9954  2c2 10947  #chash 12979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980 This theorem is referenced by:  fundmge2nop  13130  fun2dmnop0  13131  funvtxdmge2val  25691  funiedgdmge2val  25692
 Copyright terms: Public domain W3C validator