strlemor1 - Metamath Proof Explorer

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Theorem strlemor1 15796

Description: Add one element to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

**Hypotheses**
Ref	Expression
strlemor.f	⊢ (Fun ^◡^◡𝐹 ∧ dom 𝐹 ⊆ (1...𝐼))
strlemor.i	⊢ 𝐼 ∈ ℕ₀
strlemor.o	⊢ 𝐼 < 𝐽
strlemor.j	⊢ 𝐽 ∈ ℕ
strlemor.a	⊢ 𝐴 = 𝐽
strlemor1.g	⊢ 𝐺 = (𝐹 ∪ {⟨𝐴, 𝑋⟩})

**Assertion**
Ref	Expression
strlemor1	⊢ (Fun ^◡^◡𝐺 ∧ dom 𝐺 ⊆ (1...𝐽))

**Proof of Theorem strlemor1**
Step	Hyp	Ref	Expression
1		strlemor.f	. . . . . 6 ⊢ (Fun ^◡^◡𝐹 ∧ dom 𝐹 ⊆ (1...𝐼))
2	1	simpli 473	. . . . 5 ⊢ Fun ^◡^◡𝐹
3		funcnvsn 5850	. . . . 5 ⊢ Fun ^◡{⟨𝑋, 𝐽⟩}
4	2, 3	pm3.2i 470	. . . 4 ⊢ (Fun ^◡^◡𝐹 ∧ Fun ^◡{⟨𝑋, 𝐽⟩})
5		cnvcnvss 5507	. . . . . . 7 ⊢ ^◡^◡𝐹 ⊆ 𝐹
6		dmss 5245	. . . . . . 7 ⊢ (^◡^◡𝐹 ⊆ 𝐹 → dom ^◡^◡𝐹 ⊆ dom 𝐹)
7	5, 6	ax-mp 5	. . . . . 6 ⊢ dom ^◡^◡𝐹 ⊆ dom 𝐹
8		cnvcnvsn 5530	. . . . . . . . 9 ⊢ ^◡^◡{⟨𝐽, 𝑋⟩} = ^◡{⟨𝑋, 𝐽⟩}
9		cnvcnvss 5507	. . . . . . . . 9 ⊢ ^◡^◡{⟨𝐽, 𝑋⟩} ⊆ {⟨𝐽, 𝑋⟩}
10	8, 9	eqsstr3i 3599	. . . . . . . 8 ⊢ ^◡{⟨𝑋, 𝐽⟩} ⊆ {⟨𝐽, 𝑋⟩}
11		dmss 5245	. . . . . . . 8 ⊢ (^◡{⟨𝑋, 𝐽⟩} ⊆ {⟨𝐽, 𝑋⟩} → dom ^◡{⟨𝑋, 𝐽⟩} ⊆ dom {⟨𝐽, 𝑋⟩})
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ dom ^◡{⟨𝑋, 𝐽⟩} ⊆ dom {⟨𝐽, 𝑋⟩}
13		dmsnopss 5525	. . . . . . 7 ⊢ dom {⟨𝐽, 𝑋⟩} ⊆ {𝐽}
14	12, 13	sstri 3577	. . . . . 6 ⊢ dom ^◡{⟨𝑋, 𝐽⟩} ⊆ {𝐽}
15		ss2in 3802	. . . . . 6 ⊢ ((dom ^◡^◡𝐹 ⊆ dom 𝐹 ∧ dom ^◡{⟨𝑋, 𝐽⟩} ⊆ {𝐽}) → (dom ^◡^◡𝐹 ∩ dom ^◡{⟨𝑋, 𝐽⟩}) ⊆ (dom 𝐹 ∩ {𝐽}))
16	7, 14, 15	mp2an 704	. . . . 5 ⊢ (dom ^◡^◡𝐹 ∩ dom ^◡{⟨𝑋, 𝐽⟩}) ⊆ (dom 𝐹 ∩ {𝐽})
17		strlemor.o	. . . . . . . . 9 ⊢ 𝐼 < 𝐽
18		strlemor.i	. . . . . . . . . . 11 ⊢ 𝐼 ∈ ℕ₀
19	18	nn0rei 11180	. . . . . . . . . 10 ⊢ 𝐼 ∈ ℝ
20		strlemor.j	. . . . . . . . . . 11 ⊢ 𝐽 ∈ ℕ
21	20	nnrei 10906	. . . . . . . . . 10 ⊢ 𝐽 ∈ ℝ
22	19, 21	ltnlei 10037	. . . . . . . . 9 ⊢ (𝐼 < 𝐽 ↔ ¬ 𝐽 ≤ 𝐼)
23	17, 22	mpbi 219	. . . . . . . 8 ⊢ ¬ 𝐽 ≤ 𝐼
24		elfzle2 12216	. . . . . . . 8 ⊢ (𝐽 ∈ (1...𝐼) → 𝐽 ≤ 𝐼)
25	23, 24	mto 187	. . . . . . 7 ⊢ ¬ 𝐽 ∈ (1...𝐼)
26	1	simpri 477	. . . . . . . 8 ⊢ dom 𝐹 ⊆ (1...𝐼)
27	26	sseli 3564	. . . . . . 7 ⊢ (𝐽 ∈ dom 𝐹 → 𝐽 ∈ (1...𝐼))
28	25, 27	mto 187	. . . . . 6 ⊢ ¬ 𝐽 ∈ dom 𝐹
29		disjsn 4192	. . . . . 6 ⊢ ((dom 𝐹 ∩ {𝐽}) = ∅ ↔ ¬ 𝐽 ∈ dom 𝐹)
30	28, 29	mpbir 220	. . . . 5 ⊢ (dom 𝐹 ∩ {𝐽}) = ∅
31		sseq0 3927	. . . . 5 ⊢ (((dom ^◡^◡𝐹 ∩ dom ^◡{⟨𝑋, 𝐽⟩}) ⊆ (dom 𝐹 ∩ {𝐽}) ∧ (dom 𝐹 ∩ {𝐽}) = ∅) → (dom ^◡^◡𝐹 ∩ dom ^◡{⟨𝑋, 𝐽⟩}) = ∅)
32	16, 30, 31	mp2an 704	. . . 4 ⊢ (dom ^◡^◡𝐹 ∩ dom ^◡{⟨𝑋, 𝐽⟩}) = ∅
33		funun 5846	. . . 4 ⊢ (((Fun ^◡^◡𝐹 ∧ Fun ^◡{⟨𝑋, 𝐽⟩}) ∧ (dom ^◡^◡𝐹 ∩ dom ^◡{⟨𝑋, 𝐽⟩}) = ∅) → Fun (^◡^◡𝐹 ∪ ^◡{⟨𝑋, 𝐽⟩}))
34	4, 32, 33	mp2an 704	. . 3 ⊢ Fun (^◡^◡𝐹 ∪ ^◡{⟨𝑋, 𝐽⟩})
35		strlemor1.g	. . . . . . . . 9 ⊢ 𝐺 = (𝐹 ∪ {⟨𝐴, 𝑋⟩})
36		strlemor.a	. . . . . . . . . . . 12 ⊢ 𝐴 = 𝐽
37	36	opeq1i 4343	. . . . . . . . . . 11 ⊢ ⟨𝐴, 𝑋⟩ = ⟨𝐽, 𝑋⟩
38	37	sneqi 4136	. . . . . . . . . 10 ⊢ {⟨𝐴, 𝑋⟩} = {⟨𝐽, 𝑋⟩}
39	38	uneq2i 3726	. . . . . . . . 9 ⊢ (𝐹 ∪ {⟨𝐴, 𝑋⟩}) = (𝐹 ∪ {⟨𝐽, 𝑋⟩})
40	35, 39	eqtri 2632	. . . . . . . 8 ⊢ 𝐺 = (𝐹 ∪ {⟨𝐽, 𝑋⟩})
41	40	cnveqi 5219	. . . . . . 7 ⊢ ^◡𝐺 = ^◡(𝐹 ∪ {⟨𝐽, 𝑋⟩})
42		cnvun 5457	. . . . . . 7 ⊢ ^◡(𝐹 ∪ {⟨𝐽, 𝑋⟩}) = (^◡𝐹 ∪ ^◡{⟨𝐽, 𝑋⟩})
43	41, 42	eqtri 2632	. . . . . 6 ⊢ ^◡𝐺 = (^◡𝐹 ∪ ^◡{⟨𝐽, 𝑋⟩})
44	43	cnveqi 5219	. . . . 5 ⊢ ^◡^◡𝐺 = ^◡(^◡𝐹 ∪ ^◡{⟨𝐽, 𝑋⟩})
45		cnvun 5457	. . . . . 6 ⊢ ^◡(^◡𝐹 ∪ ^◡{⟨𝐽, 𝑋⟩}) = (^◡^◡𝐹 ∪ ^◡^◡{⟨𝐽, 𝑋⟩})
46	8	uneq2i 3726	. . . . . 6 ⊢ (^◡^◡𝐹 ∪ ^◡^◡{⟨𝐽, 𝑋⟩}) = (^◡^◡𝐹 ∪ ^◡{⟨𝑋, 𝐽⟩})
47	45, 46	eqtri 2632	. . . . 5 ⊢ ^◡(^◡𝐹 ∪ ^◡{⟨𝐽, 𝑋⟩}) = (^◡^◡𝐹 ∪ ^◡{⟨𝑋, 𝐽⟩})
48	44, 47	eqtri 2632	. . . 4 ⊢ ^◡^◡𝐺 = (^◡^◡𝐹 ∪ ^◡{⟨𝑋, 𝐽⟩})
49	48	funeqi 5824	. . 3 ⊢ (Fun ^◡^◡𝐺 ↔ Fun (^◡^◡𝐹 ∪ ^◡{⟨𝑋, 𝐽⟩}))
50	34, 49	mpbir 220	. 2 ⊢ Fun ^◡^◡𝐺
51	40	dmeqi 5247	. . . 4 ⊢ dom 𝐺 = dom (𝐹 ∪ {⟨𝐽, 𝑋⟩})
52		dmun 5253	. . . 4 ⊢ dom (𝐹 ∪ {⟨𝐽, 𝑋⟩}) = (dom 𝐹 ∪ dom {⟨𝐽, 𝑋⟩})
53	51, 52	eqtri 2632	. . 3 ⊢ dom 𝐺 = (dom 𝐹 ∪ dom {⟨𝐽, 𝑋⟩})
54	18	nn0zi 11279	. . . . . . 7 ⊢ 𝐼 ∈ ℤ
55	20	nnzi 11278	. . . . . . 7 ⊢ 𝐽 ∈ ℤ
56	19, 21, 17	ltleii 10039	. . . . . . 7 ⊢ 𝐼 ≤ 𝐽
57		eluz2 11569	. . . . . . 7 ⊢ (𝐽 ∈ (ℤ_≥‘𝐼) ↔ (𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ 𝐼 ≤ 𝐽))
58	54, 55, 56, 57	mpbir3an 1237	. . . . . 6 ⊢ 𝐽 ∈ (ℤ_≥‘𝐼)
59		fzss2 12252	. . . . . 6 ⊢ (𝐽 ∈ (ℤ_≥‘𝐼) → (1...𝐼) ⊆ (1...𝐽))
60	58, 59	ax-mp 5	. . . . 5 ⊢ (1...𝐼) ⊆ (1...𝐽)
61	26, 60	sstri 3577	. . . 4 ⊢ dom 𝐹 ⊆ (1...𝐽)
62		elfz1end 12242	. . . . . . 7 ⊢ (𝐽 ∈ ℕ ↔ 𝐽 ∈ (1...𝐽))
63	20, 62	mpbi 219	. . . . . 6 ⊢ 𝐽 ∈ (1...𝐽)
64		snssi 4280	. . . . . 6 ⊢ (𝐽 ∈ (1...𝐽) → {𝐽} ⊆ (1...𝐽))
65	63, 64	ax-mp 5	. . . . 5 ⊢ {𝐽} ⊆ (1...𝐽)
66	13, 65	sstri 3577	. . . 4 ⊢ dom {⟨𝐽, 𝑋⟩} ⊆ (1...𝐽)
67	61, 66	unssi 3750	. . 3 ⊢ (dom 𝐹 ∪ dom {⟨𝐽, 𝑋⟩}) ⊆ (1...𝐽)
68	53, 67	eqsstri 3598	. 2 ⊢ dom 𝐺 ⊆ (1...𝐽)
69	50, 68	pm3.2i 470	1 ⊢ (Fun ^◡^◡𝐺 ∧ dom 𝐺 ⊆ (1...𝐽))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 ⟨cop 4131 class class class wbr 4583 ^◡ccnv 5037 dom cdm 5038 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 1c1 9816 < clt 9953 ≤ cle 9954 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198

This theorem is referenced by: strlemor2 15797 strlemor3 15798

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