Theorem ssceq 16967

Description: The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)

Assertion

Ref	Expression
ssceq	⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐴 = 𝐵)

Proof of Theorem ssceq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 475	. . . . . 6 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐴 ⊆cat 𝐵)
2		eqidd 2774	. . . . . 6 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → dom dom 𝐴 = dom dom 𝐴)
3	1, 2	sscfn1 16958	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4		simpr 477	. . . . . 6 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐵 ⊆cat 𝐴)
5		eqidd 2774	. . . . . 6 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → dom dom 𝐵 = dom dom 𝐵)
6	4, 5	sscfn1 16958	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
7	3, 6, 1	ssc1 16962	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → dom dom 𝐴 ⊆ dom dom 𝐵)
8	6, 3, 4	ssc1 16962	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → dom dom 𝐵 ⊆ dom dom 𝐴)
9	7, 8	eqssd 3870	. . 3 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → dom dom 𝐴 = dom dom 𝐵)
10	9	sqxpeqd 5436	. 2 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → (dom dom 𝐴 × dom dom 𝐴) = (dom dom 𝐵 × dom dom 𝐵))
11	3	adantr 473	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
12	1	adantr 473	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐴 ⊆cat 𝐵)
13		simprl 759	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
14		simprr 761	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐴)
15	11, 12, 13, 14	ssc2 16963	. . . 4 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
16	6	adantr 473	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
17	4	adantr 473	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐵 ⊆cat 𝐴)
18	7	adantr 473	. . . . . 6 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → dom dom 𝐴 ⊆ dom dom 𝐵)
19	18, 13	sseldd 3854	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
20	18, 14	sseldd 3854	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
21	16, 17, 19, 20	ssc2 16963	. . . 4 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐴𝑦))
22	15, 21	eqssd 3870	. . 3 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) = (𝑥𝐵𝑦))
23	22	ralrimivva 3136	. 2 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → ∀𝑥 ∈ dom dom 𝐴∀𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) = (𝑥𝐵𝑦))
24		eqfnov 7095	. . 3 ⊢ ((𝐴 Fn (dom dom 𝐴 × dom dom 𝐴) ∧ 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵)) → (𝐴 = 𝐵 ↔ ((dom dom 𝐴 × dom dom 𝐴) = (dom dom 𝐵 × dom dom 𝐵) ∧ ∀𝑥 ∈ dom dom 𝐴∀𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) = (𝑥𝐵𝑦))))
25	3, 6, 24	syl2anc 576	. 2 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → (𝐴 = 𝐵 ↔ ((dom dom 𝐴 × dom dom 𝐴) = (dom dom 𝐵 × dom dom 𝐵) ∧ ∀𝑥 ∈ dom dom 𝐴∀𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) = (𝑥𝐵𝑦))))
26	10, 23, 25	mpbir2and 701	1 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508   ∈ wcel 2051  ∀wral 3083   ⊆ wss 3824   class class class wbr 4926   × cxp 5402  dom cdm 5404   Fn wfn 6181  (class class class)co 6975   ⊆_cat cssc 16948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-ov 6978  df-ixp 8259  df-ssc 16951

This theorem is referenced by: (None)