Theorem actfunsnrndisj 34270

Description: The action 𝐹 of extending function from 𝐵 to 𝐶 with new values at point 𝐼 yields different functions. (Contributed by Thierry Arnoux, 9-Dec-2021.)

Hypotheses

Ref	Expression
actfunsn.1	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐴 ⊆ (𝐶 ↑m 𝐵))
actfunsn.2	⊢ (𝜑 → 𝐶 ∈ V)
actfunsn.3	⊢ (𝜑 → 𝐼 ∈ 𝑉)
actfunsn.4	⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵)
actfunsn.5	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))

Assertion

Ref	Expression
actfunsnrndisj	⊢ (𝜑 → Disj 𝑘 ∈ 𝐶 ran 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑘,𝐼,𝑥   𝜑,𝑘

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑘)   𝐵(𝑥,𝑘)   𝐶(𝑥,𝑘)   𝐹(𝑥,𝑘)   𝑉(𝑥,𝑘)

Proof of Theorem actfunsnrndisj

Dummy variables 𝑧 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 483	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
2	1	fveq1d 6904	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓‘𝐼) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼))
3		actfunsn.1	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐴 ⊆ (𝐶 ↑m 𝐵))
4	3	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆ (𝐶 ↑m 𝐵))
5		simpr 483	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
6	4, 5	sseldd 3983	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝐶 ↑m 𝐵))
7		elmapfn 8890	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝐶 ↑m 𝐵) → 𝑧 Fn 𝐵)
8	6, 7	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝑧 Fn 𝐵)
9		actfunsn.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
10	9	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝐼 ∈ 𝑉)
11		simpllr 774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝑘 ∈ 𝐶)
12		fnsng 6610	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
13	10, 11, 12	syl2anc 582	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
14		actfunsn.4	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵)
15		disjsn 4720	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ 𝐵)
16	14, 15	sylibr 233	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
17	16	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → (𝐵 ∩ {𝐼}) = ∅)
18		snidg 4667	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼})
19	10, 18	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝐼 ∈ {𝐼})
20		fvun2 6995	. . . . . . . . 9 ⊢ ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ ((𝐵 ∩ {𝐼}) = ∅ ∧ 𝐼 ∈ {𝐼})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
21	8, 13, 17, 19, 20	syl112anc 1371	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
22		fvsng 7195	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
23	10, 11, 22	syl2anc 582	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
24	21, 23	eqtrd 2768	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
25	24	adantr 479	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
26	2, 25	eqtrd 2768	. . . . 5 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓‘𝐼) = 𝑘)
27		simpr 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) → 𝑓 ∈ ran 𝐹)
28		actfunsn.5	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))
29		uneq1 4157	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
30	29	cbvmptv 5265	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
31	28, 30	eqtri 2756	. . . . . . 7 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32		vex 3477	. . . . . . . 8 ⊢ 𝑧 ∈ V
33		snex 5437	. . . . . . . 8 ⊢ {⟨𝐼, 𝑘⟩} ∈ V
34	32, 33	unex 7754	. . . . . . 7 ⊢ (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
35	31, 34	elrnmpti 5966	. . . . . 6 ⊢ (𝑓 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
36	27, 35	sylib 217	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) → ∃𝑧 ∈ 𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
37	26, 36	r19.29a 3159	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) → (𝑓‘𝐼) = 𝑘)
38	37	ralrimiva 3143	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → ∀𝑓 ∈ ran 𝐹(𝑓‘𝐼) = 𝑘)
39	38	ralrimiva 3143	. 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐶 ∀𝑓 ∈ ran 𝐹(𝑓‘𝐼) = 𝑘)
40		invdisj 5136	. 2 ⊢ (∀𝑘 ∈ 𝐶 ∀𝑓 ∈ ran 𝐹(𝑓‘𝐼) = 𝑘 → Disj 𝑘 ∈ 𝐶 ran 𝐹)
41	39, 40	syl 17	1 ⊢ (𝜑 → Disj 𝑘 ∈ 𝐶 ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ∃wrex 3067  Vcvv 3473   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ∅c0 4326  {csn 4632  ⟨cop 4638  Disj wdisj 5117   ↦ cmpt 5235  ran crn 5683   Fn wfn 6548  ‘cfv 6553  (class class class)co 7426   ↑_m cmap 8851

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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-disj 5118  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-map 8853

This theorem is referenced by:  breprexplema  34295