Theorem fundm2domnop0 11220

Description: A function with a domain containing (at least) two different elements is not an ordered pair. This theorem (which requires that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 13225. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Proof shortened by AV, 15-Nov-2021.)

Assertion

Ref	Expression
fundm2domnop0	⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2o ≼ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

Proof of Theorem fundm2domnop0

Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2dom 7046	. . 3 ⊢ (2o ≼ dom 𝐺 → ∃𝑎 ∈ dom 𝐺∃𝑏 ∈ dom 𝐺 ¬ 𝑎 = 𝑏)
2		elvv 4812	. . . . . . . 8 ⊢ (𝐺 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐺 = ⟨𝑥, 𝑦⟩)
3		difeq1 3330	. . . . . . . . . . . . 13 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (𝐺 ∖ {∅}) = (⟨𝑥, 𝑦⟩ ∖ {∅}))
4	3	funeqd 5374	. . . . . . . . . . . 12 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (𝐺 ∖ {∅}) ↔ Fun (⟨𝑥, 𝑦⟩ ∖ {∅})))
5		opwo0id 4365	. . . . . . . . . . . . . . 15 ⊢ ⟨𝑥, 𝑦⟩ = (⟨𝑥, 𝑦⟩ ∖ {∅})
6	5	eqcomi 2236	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑦⟩ ∖ {∅}) = ⟨𝑥, 𝑦⟩
7	6	funeqi 5373	. . . . . . . . . . . . 13 ⊢ (Fun (⟨𝑥, 𝑦⟩ ∖ {∅}) ↔ Fun ⟨𝑥, 𝑦⟩)
8		dmeq 4956	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → dom 𝐺 = dom ⟨𝑥, 𝑦⟩)
9	8	eleq2d 2302	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (𝑎 ∈ dom 𝐺 ↔ 𝑎 ∈ dom ⟨𝑥, 𝑦⟩))
10	8	eleq2d 2302	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (𝑏 ∈ dom 𝐺 ↔ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩))
11	9, 10	anbi12d 473	. . . . . . . . . . . . . . 15 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ↔ (𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩)))
12		eqid 2232	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑦⟩
13		vex 2816	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
14		vex 2816	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
15	12, 13, 14	funopdmsn 5864	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun ⟨𝑥, 𝑦⟩ ∧ 𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩) → 𝑎 = 𝑏)
16	15	3expb 1231	. . . . . . . . . . . . . . . 16 ⊢ ((Fun ⟨𝑥, 𝑦⟩ ∧ (𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩)) → 𝑎 = 𝑏)
17	16	expcom 116	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ dom ⟨𝑥, 𝑦⟩ ∧ 𝑏 ∈ dom ⟨𝑥, 𝑦⟩) → (Fun ⟨𝑥, 𝑦⟩ → 𝑎 = 𝑏))
18	11, 17	biimtrdi 163	. . . . . . . . . . . . . 14 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → (Fun ⟨𝑥, 𝑦⟩ → 𝑎 = 𝑏)))
19	18	com23 78	. . . . . . . . . . . . 13 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → 𝑎 = 𝑏)))
20	7, 19	biimtrid 152	. . . . . . . . . . . 12 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (⟨𝑥, 𝑦⟩ ∖ {∅}) → ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → 𝑎 = 𝑏)))
21	4, 20	sylbid 150	. . . . . . . . . . 11 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun (𝐺 ∖ {∅}) → ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → 𝑎 = 𝑏)))
22	21	impcomd 255	. . . . . . . . . 10 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → 𝑎 = 𝑏))
23	22	exlimivv 1946	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → (((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → 𝑎 = 𝑏))
24	23	com12 30	. . . . . . . 8 ⊢ (((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → (∃𝑥∃𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → 𝑎 = 𝑏))
25	2, 24	biimtrid 152	. . . . . . 7 ⊢ (((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → (𝐺 ∈ (V × V) → 𝑎 = 𝑏))
26	25	con3d 636	. . . . . 6 ⊢ (((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) ∧ Fun (𝐺 ∖ {∅})) → (¬ 𝑎 = 𝑏 → ¬ 𝐺 ∈ (V × V)))
27	26	ex 115	. . . . 5 ⊢ ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → (Fun (𝐺 ∖ {∅}) → (¬ 𝑎 = 𝑏 → ¬ 𝐺 ∈ (V × V))))
28	27	com23 78	. . . 4 ⊢ ((𝑎 ∈ dom 𝐺 ∧ 𝑏 ∈ dom 𝐺) → (¬ 𝑎 = 𝑏 → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V))))
29	28	rexlimivv 2666	. . 3 ⊢ (∃𝑎 ∈ dom 𝐺∃𝑏 ∈ dom 𝐺 ¬ 𝑎 = 𝑏 → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V)))
30	1, 29	syl 14	. 2 ⊢ (2o ≼ dom 𝐺 → (Fun (𝐺 ∖ {∅}) → ¬ 𝐺 ∈ (V × V)))
31	30	impcom 125	1 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2o ≼ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∃wrex 2521  Vcvv 2813   ∖ cdif 3208  ∅c0 3508  {csn 3689  ⟨cop 3692   class class class wbr 4109   × cxp 4747  dom cdm 4749  Fun wfun 5346  2_oc2o 6641   ≼ cdom 6974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fv 5360  df-1o 6647  df-2o 6648  df-dom 6977

This theorem is referenced by:  fundm2domnop  11221  fun2dmnop0  11222  funvtxdm2domval  16024  funiedgdm2domval  16025