Theorem ssceq 17883

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 482	. . . . . 6 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐴 ⊆cat 𝐵)
2		eqidd 2738	. . . . . 6 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → dom dom 𝐴 = dom dom 𝐴)
3	1, 2	sscfn1 17874	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4		simpr 484	. . . . . 6 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐵 ⊆cat 𝐴)
5		eqidd 2738	. . . . . 6 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → dom dom 𝐵 = dom dom 𝐵)
6	4, 5	sscfn1 17874	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
7	3, 6, 1	ssc1 17878	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → dom dom 𝐴 ⊆ dom dom 𝐵)
8	6, 3, 4	ssc1 17878	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → dom dom 𝐵 ⊆ dom dom 𝐴)
9	7, 8	eqssd 4016	. . 3 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → dom dom 𝐴 = dom dom 𝐵)
10	9	sqxpeqd 5725	. 2 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → (dom dom 𝐴 × dom dom 𝐴) = (dom dom 𝐵 × dom dom 𝐵))
11	3	adantr 480	. . . . 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 480	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐴 ⊆cat 𝐵)
13		simprl 771	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
14		simprr 773	. . . . 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 17879	. . . 4 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
16	6	adantr 480	. . . . 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 480	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐵 ⊆cat 𝐴)
18	7	adantr 480	. . . . . 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 3999	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
20	18, 14	sseldd 3999	. . . . 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 17879	. . . 4 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐴𝑦))
22	15, 21	eqssd 4016	. . 3 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) = (𝑥𝐵𝑦))
23	22	ralrimivva 3202	. 2 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → ∀𝑥 ∈ dom dom 𝐴∀𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) = (𝑥𝐵𝑦))
24		eqfnov 7569	. . 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 584	. 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 713	1 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3966   class class class wbr 5151   × cxp 5691  dom cdm 5693   Fn wfn 6564  (class class class)co 7438   ⊆_cat cssc 17864

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-ov 7441  df-ixp 8946  df-ssc 17867

This theorem is referenced by: (None)