Theorem fundmge2nop0 14401

Description: A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fundmge2nop 14402 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 17054. (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.

Step	Hyp	Ref	Expression
1		dmexg 7826	. . . . . 6 ⊢ (𝐺 ∈ V → dom 𝐺 ∈ V)
2		hashge2el2dif 14379	. . . . . . 7 ⊢ ((dom 𝐺 ∈ V ∧ 2 ≤ (♯‘dom 𝐺)) → ∃𝑎 ∈ dom 𝐺∃𝑏 ∈ dom 𝐺 𝑎 ≠ 𝑏)
3	2	ex 412	. . . . . 6 ⊢ (dom 𝐺 ∈ V → (2 ≤ (♯‘dom 𝐺) → ∃𝑎 ∈ dom 𝐺∃𝑏 ∈ dom 𝐺 𝑎 ≠ 𝑏))
4	1, 3	syl 17	. . . . 5 ⊢ (𝐺 ∈ V → (2 ≤ (♯‘dom 𝐺) → ∃𝑎 ∈ dom 𝐺∃𝑏 ∈ dom 𝐺 𝑎 ≠ 𝑏))
5		df-ne 2927	. . . . . . 7 ⊢ (𝑎 ≠ 𝑏 ↔ ¬ 𝑎 = 𝑏)
6		elvv 5689	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐺 = ⟨𝑥, 𝑦⟩)
7		difeq1 4067	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (𝐺 ∖ {∅}) = (⟨𝑥, 𝑦⟩ ∖ {∅}))
8	7	funeqd 6499	. . . . . . . . . . . . . . 15 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (𝐺 ∖ {∅}) ↔ Fun (⟨𝑥, 𝑦⟩ ∖ {∅})))
9		opwo0id 5435	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝑥, 𝑦⟩ = (⟨𝑥, 𝑦⟩ ∖ {∅})
10	9	eqcomi 2739	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑥, 𝑦⟩ ∖ {∅}) = ⟨𝑥, 𝑦⟩
11	10	funeqi 6498	. . . . . . . . . . . . . . . 16 ⊢ (Fun (⟨𝑥, 𝑦⟩ ∖ {∅}) ↔ Fun ⟨𝑥, 𝑦⟩)
12		dmeq 5841	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → dom 𝐺 = dom ⟨𝑥, 𝑦⟩)
13	12	eleq2d 2815	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (𝑎 ∈ dom 𝐺 ↔ 𝑎 ∈ dom ⟨𝑥, 𝑦⟩))
14	12	eleq2d 2815	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (𝑏 ∈ dom 𝐺 ↔ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩))
15	13, 14	anbi12d 632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ↔ (𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩)))
16		eqid 2730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑦⟩
17		vex 3438	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ∈ V
18		vex 3438	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑦 ∈ V
19	16, 17, 18	funopdmsn 7078	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun ⟨𝑥, 𝑦⟩ ∧ 𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩) → 𝑎 = 𝑏)
20	19	3expb 1120	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun ⟨𝑥, 𝑦⟩ ∧ (𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩)) → 𝑎 = 𝑏)
21	20	expcom 413	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩) → (Fun ⟨𝑥, 𝑦⟩ → 𝑎 = 𝑏))
22	15, 21	biimtrdi 253	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → (Fun ⟨𝑥, 𝑦⟩ → 𝑎 = 𝑏)))
23	22	com23 86	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → 𝑎 = 𝑏)))
24	11, 23	biimtrid 242	. . . . . . . . . . . . . . 15 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (⟨𝑥, 𝑦⟩ ∖ {∅}) → ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → 𝑎 = 𝑏)))
25	8, 24	sylbid 240	. . . . . . . . . . . . . 14 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (𝐺 ∖ {∅}) → ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → 𝑎 = 𝑏)))
26	25	impcomd 411	. . . . . . . . . . . . 13 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → 𝑎 = 𝑏))
27	26	exlimivv 1933	. . . . . . . . . . . 12 ⊢ (∃𝑥∃𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → (((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → 𝑎 = 𝑏))
28	27	com12 32	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → (∃𝑥∃𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → 𝑎 = 𝑏))
29	6, 28	biimtrid 242	. . . . . . . . . 10 ⊢ (((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → (𝐺 ∈ (V × V) → 𝑎 = 𝑏))
30	29	con3d 152	. . . . . . . . 9 ⊢ (((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → (¬ 𝑎 = 𝑏 → ¬ 𝐺 ∈ (V × V)))
31	30	ex 412	. . . . . . . 8 ⊢ ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → (Fun (𝐺 ∖ {∅}) → (¬ 𝑎 = 𝑏 → ¬ 𝐺 ∈ (V × V))))
32	31	com23 86	. . . . . . 7 ⊢ ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → (¬ 𝑎 = 𝑏 → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V))))
33	5, 32	biimtrid 242	. . . . . 6 ⊢ ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → (𝑎 ≠ 𝑏 → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V))))
34	33	rexlimivv 3172	. . . . 5 ⊢ (∃𝑎 ∈ dom 𝐺∃𝑏 ∈ dom 𝐺 𝑎 ≠ 𝑏 → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V)))
35	4, 34	syl6 35	. . . 4 ⊢ (𝐺 ∈ V → (2 ≤ (♯‘dom 𝐺) → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V))))
36	35	com13 88	. . 3 ⊢ (Fun (𝐺 ∖ {∅}) → (2 ≤ (♯‘dom 𝐺) → (𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))))
37	36	imp 406	. 2 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (𝐺 ∈ V → ¬ 𝐺 ∈ (V × V)))
38		prcnel 3460	. 2 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
39	37, 38	pm2.61d1 180	1 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → ¬ 𝐺 ∈ (V × V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2110   ≠ wne 2926  ∃wrex 3054  Vcvv 3434   ∖ cdif 3897  ∅c0 4281  {csn 4574  ⟨cop 4580   class class class wbr 5089   × cxp 5612  dom cdm 5614  Fun wfun 6471  ‘cfv 6477   ≤ cle 11139  2c2 12172  ♯chash 14229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-n0 12374  df-xnn0 12447  df-z 12461  df-uz 12725  df-fz 13400  df-hash 14230

This theorem is referenced by:  fundmge2nop  14402  fun2dmnop0  14403  funvtxdmge2val  28982  funiedgdmge2val  28983