Theorem actfunsnrndisj 34896

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 6869	. . . . . 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 736	. . . . . . . . . . 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 3937	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝐶 ↑m 𝐵))
7		elmapfn 8846	. . . . . . . . . 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 740	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝐼 ∈ 𝑉)
11		simpllr 785	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝑘 ∈ 𝐶)
12		fnsng 6573	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
13	10, 11, 12	syl2anc 593	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
14		actfunsn.4	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵)
15		disjsn 4670	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ 𝐵)
16	14, 15	sylibr 236	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
17	16	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → (𝐵 ∩ {𝐼}) = ∅)
18		snidg 4619	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼})
19	10, 18	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → 𝐼 ∈ {𝐼})
20		fvun2 6959	. . . . . . . . 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 1393	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
22		fvsng 7164	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
23	10, 11, 22	syl2anc 593	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
24	21, 23	eqtrd 2797	. . . . . . 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 2797	. . . . 5 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓‘𝐼) = 𝑘)
27		actfunsn.5	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))
28		uneq1 4114	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
29	28	cbvmptv 5204	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
30	27, 29	eqtri 2785	. . . . . . 7 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
31		vex 3458	. . . . . . . 8 ⊢ 𝑧 ∈ V
32		snex 5396	. . . . . . . 8 ⊢ {⟨𝐼, 𝑘⟩} ∈ V
33	31, 32	unex 7727	. . . . . . 7 ⊢ (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
34	30, 33	elrnmpti 5938	. . . . . 6 ⊢ (𝑓 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
35	34	bilani 508	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) → ∃𝑧 ∈ 𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
36	26, 35	r19.29a 3170	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑓 ∈ ran 𝐹) → (𝑓‘𝐼) = 𝑘)
37	36	ralrimiva 3154	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → ∀𝑓 ∈ ran 𝐹(𝑓‘𝐼) = 𝑘)
38	37	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐶 ∀𝑓 ∈ ran 𝐹(𝑓‘𝐼) = 𝑘)
39		invdisj 5086	. 2 ⊢ (∀𝑘 ∈ 𝐶 ∀𝑓 ∈ ran 𝐹(𝑓‘𝐼) = 𝑘 → Disj 𝑘 ∈ 𝐶 ran 𝐹)
40	38, 39	syl 17	1 ⊢ (𝜑 → Disj 𝑘 ∈ 𝐶 ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  Vcvv 3454   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4582  ⟨cop 4588  Disj wdisj 5067   ↦ cmpt 5181  ran crn 5648   Fn wfn 6516  ‘cfv 6521  (class class class)co 7396   ↑_m cmap 8808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-disj 5068  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810

This theorem is referenced by:  breprexplema  34921