Theorem weisoeq2 6769

Description: Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 7319. (Contributed by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
weisoeq2	⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)

Proof of Theorem weisoeq2

Step	Hyp	Ref	Expression
1		isocnv 6743	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2		isocnv 6743	. . . 4 ⊢ (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐺 Isom 𝑆, 𝑅 (𝐵, 𝐴))
3	1, 2	anim12i 591	. . 3 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → (◡𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴) ∧ ◡𝐺 Isom 𝑆, 𝑅 (𝐵, 𝐴)))
4		weisoeq 6768	. . 3 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (◡𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴) ∧ ◡𝐺 Isom 𝑆, 𝑅 (𝐵, 𝐴))) → ◡𝐹 = ◡𝐺)
5	3, 4	sylan2 492	. 2 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → ◡𝐹 = ◡𝐺)
6		simprl 811	. . . 4 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
7		isof1o 6736	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
8		f1orel 6301	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
9	6, 7, 8	3syl 18	. . 3 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → Rel 𝐹)
10		simprr 813	. . . 4 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))
11		isof1o 6736	. . . 4 ⊢ (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐺:𝐴–1-1-onto→𝐵)
12		f1orel 6301	. . . 4 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → Rel 𝐺)
13	10, 11, 12	3syl 18	. . 3 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → Rel 𝐺)
14		cnveqb 5748	. . 3 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → (𝐹 = 𝐺 ↔ ◡𝐹 = ◡𝐺))
15	9, 13, 14	syl2anc 696	. 2 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝐹 = 𝐺 ↔ ◡𝐹 = ◡𝐺))
16	5, 15	mpbird 247	1 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   Se wse 5223   We wwe 5224  ^◡ccnv 5265  Rel wrel 5271  –1-1-onto→wf1o 6048   Isom wiso 6050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058

This theorem is referenced by:  wemoiso2  7319