Theorem actfunsnrndisj 31986

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 488	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
2	1	fveq1d 6647	. . . . . 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 725	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆ (𝐶 ↑m 𝐵))
5		simpr 488	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
6	4, 5	sseldd 3916	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝐶 ↑m 𝐵))
7		elmapfn 8412	. . . . . . . . . 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 729	. . . . . . . . . 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 6376	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
13	10, 11, 12	syl2anc 587	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
14		actfunsn.4	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵)
15		disjsn 4607	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ 𝐵)
16	14, 15	sylibr 237	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
17	16	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → (𝐵 ∩ {𝐼}) = ∅)
18		snidg 4559	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼})
19	10, 18	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝐼 ∈ {𝐼})
20		fvun2 6730	. . . . . . . . 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 6919	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
23	10, 11, 22	syl2anc 587	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
24	21, 23	eqtrd 2833	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
25	24	adantr 484	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
26	2, 25	eqtrd 2833	. . . . 5 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓‘𝐼) = 𝑘)
27		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) → 𝑓 ∈ ran 𝐹)
28		actfunsn.5	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))
29		uneq1 4083	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
30	29	cbvmptv 5133	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
31	28, 30	eqtri 2821	. . . . . . 7 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32		vex 3444	. . . . . . . 8 ⊢ 𝑧 ∈ V
33		snex 5297	. . . . . . . 8 ⊢ {⟨𝐼, 𝑘⟩} ∈ V
34	32, 33	unex 7449	. . . . . . 7 ⊢ (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
35	31, 34	elrnmpti 5796	. . . . . 6 ⊢ (𝑓 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
36	27, 35	sylib 221	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) → ∃𝑧 ∈ 𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
37	26, 36	r19.29a 3248	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) → (𝑓‘𝐼) = 𝑘)
38	37	ralrimiva 3149	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → ∀𝑓 ∈ ran 𝐹(𝑓‘𝐼) = 𝑘)
39	38	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐶 ∀𝑓 ∈ ran 𝐹(𝑓‘𝐼) = 𝑘)
40		invdisj 5014	. 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 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {csn 4525  ⟨cop 4531  Disj wdisj 4995   ↦ cmpt 5110  ran crn 5520   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-disj 4996  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391

This theorem is referenced by:  breprexplema  32011