Theorem actfunsnrndisj 34621

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 484	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
2	1	fveq1d 6907	. . . . . 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 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆ (𝐶 ↑m 𝐵))
5		simpr 484	. . . . . . . . . . 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 8906	. . . . . . . . . 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 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝐼 ∈ 𝑉)
11		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝑘 ∈ 𝐶)
12		fnsng 6617	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
13	10, 11, 12	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
14		actfunsn.4	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵)
15		disjsn 4710	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ 𝐵)
16	14, 15	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
17	16	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → (𝐵 ∩ {𝐼}) = ∅)
18		snidg 4659	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼})
19	10, 18	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝐼 ∈ {𝐼})
20		fvun2 7000	. . . . . . . . 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 1375	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
22		fvsng 7201	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
23	10, 11, 22	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
24	21, 23	eqtrd 2776	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
25	24	adantr 480	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
26	2, 25	eqtrd 2776	. . . . 5 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓‘𝐼) = 𝑘)
27		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) → 𝑓 ∈ ran 𝐹)
28		actfunsn.5	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))
29		uneq1 4160	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
30	29	cbvmptv 5254	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
31	28, 30	eqtri 2764	. . . . . . 7 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32		vex 3483	. . . . . . . 8 ⊢ 𝑧 ∈ V
33		snex 5435	. . . . . . . 8 ⊢ {⟨𝐼, 𝑘⟩} ∈ V
34	32, 33	unex 7765	. . . . . . 7 ⊢ (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
35	31, 34	elrnmpti 5972	. . . . . 6 ⊢ (𝑓 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
36	27, 35	sylib 218	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) → ∃𝑧 ∈ 𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
37	26, 36	r19.29a 3161	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) → (𝑓‘𝐼) = 𝑘)
38	37	ralrimiva 3145	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → ∀𝑓 ∈ ran 𝐹(𝑓‘𝐼) = 𝑘)
39	38	ralrimiva 3145	. 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐶 ∀𝑓 ∈ ran 𝐹(𝑓‘𝐼) = 𝑘)
40		invdisj 5128	. 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 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ∪ cun 3948   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  {csn 4625  ⟨cop 4631  Disj wdisj 5109   ↦ cmpt 5224  ran crn 5685   Fn wfn 6555  ‘cfv 6560  (class class class)co 7432   ↑_m cmap 8867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-disj 5110  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869

This theorem is referenced by:  breprexplema  34646